wtf

Why are you saying so? It's not true.

Americans (US citizens, usually poorly educated)...

How were we to interpret this remark?

On the other hand, posting that you have a degree from Istanbul and therefore by inference you must be right is seriously ludicrous.

Sensei is not the poster from Istanbul, that's Blue89. Sensei is the poster who doesn't like Americans.

What is the domain of your function? In your example of setting x = 0, 1, 2, 3 ... are those the inputs or the outputs?

 

If you mean you have some function [math]f : \mathbb N \rightarrow \mathbb N[/math] with [math]f(0) = 0[/math], [math]f(1) = 1[/math], etc., then there's an easy way to get unique outputs for any number of variables.

 

For example given two positive integers [math]m, n[/math] we can map that pair to [math]2^m 3^n[/math].

 

Likewise given any finite set of [math]n[/math] positive integer input variables [math]n_i[/math], we can map them to [math]\displaystyle \Pi_{i=1}^n p_i^{n_i}[/math] where [math]p_i[/math] is the [math]i[/math]-th prime.

 

Uniqueness of the output is guaranteed by the fundamental theorem of arithmetic. Do you see how this works?

 

Is this what you are asking?

I have another idle thought.

 

Bringing in probability, say that each person has one "shot" that will kill someone with probability 1/2. I would not be surprised if every variation of rules reduces to the scenario where each person shoots themselves. Half will live, half will die. I don't see why this would be any different than any other scenario, under the assumption that if you are killed you can't kill someone (after you get killed).

I think the idea is that some people get killed before they get a chance to kill.

Tricky little problem then, just to figure out what the rules are. For example, it seems reasonable to say that "Nobody can kill anyone if they themselves have been killed."

 

That leads to a logical contradiction. Say you have ten people, 1 through 10, and person n kills person n+1, so everyone is dead except 1.

 

Ok, think about 10. He was killed by 9. Ok, fine, but ... 9 was killed by 8. So by our rule, 9 did not kill 10.

 

Working backwards, nobody can get killed. But if nobody gets killed, then somebody MUST get killed. Contradiction.

 

We need to formalize the rules of this game, and that seems trickier than it seems.

 

(ps) -- If we introduce time, and say that only one person at a time can kill; and we weaken our rule to "Nobody can kill anyone if they have ALREADY been killed," this might be a workable model of the problem

 

In other words if they're going to get killed in the FUTURE, then they can kill someone in the present.

I don't follow your logic. If everyone kills one person and (presumably) nobody can be killed twice, then everyone must get killed. How do you figure otherwise?

 

If the population is 10, say, and each person kills one person, then the total number of people killed is ten. There's no probability distribution, no pairing. Under the hypothesis that everyone in the world kills one person, it's not possible for anyone to survive. They can each kill themselves, or they can pair up and kill each other, or person n can kill person n+1 and the last person kills the first (cyclic homicide). No matter how you arrange this, everyone gets killed.

 

Can you walk through your idea that it's possible for anyone to not be killed?

 

Do you mean perhaps that each person has some probability of killing one person? That's a totally different question than what you asked.

You gave the correct word (red). The ellipsis was invented for just that reason. Since there is no largest integer, no one can write it. Sequences are not laid down instantaneously, so it's a real question as to how long to form one that has no end. .

Do you believe in functions? For example if we say that f(x) = x^2, do you believe in f? Because an infinite sequence is just a function whose domain is the natural numbers. All the mappings from elements of the domain to elements of the range exist at the same time, as mathematical sets. There is no time involved.

 

Of if you like you can think of a function as a machine. Input a value and get out a value. For the squaring function, input 5 and the machine outputs 25. If you have a sequence, it's just a function where you input n and the machine outputs the n-th element of the sequence.

 

If you read below the tree, it says each sequence corresponds to a unique path.

You're right, but then I don't understand what is the point of showing the tree. The set of nodes is countable, but the set of paths is uncountable.

You begin at L, toss a coin, which selects 0 or 1, move 1 segment/branch, repeat...

The choice is always the same, and it's always from L!

No, an infinite sequence is NEVER a node in the tree. It's obvious that any node in the tree represents a FINITE sequence.

 

An infinite sequence is a PATH through the tree, and not a node. If you don't see this you don't understand the infinite binary tree.

Phyti, I found a couple of problems in your paper.

 

1) Your tree does not contain all binary sequences. For example, what node of the tree contains the sequence 101010101..., that is, alternating 1's and 0s'? The solution is that real numbers are represented as paths in the infinite binary tree, not nodes. Although there are only countably many nodes in the tree, there are uncountably many paths.

 

2) You wrote: "For large values of k the "end" of the list effectively accelerates into the future ..."

 

This of course is incoherent. Does the sequence of natural numbers 1, 2, 3, ... "accelerate into the future?" This is simply meaningless.

Can you say how you are able to prove something for all n by a process of number crunching? For example you can't prove the there are infinitely many primes using a spreadsheet. You have to come up with a clever idea that handles all the cases at once.

 

Anyway really can't help you on the intellectual property issue, if priority is really your concern you should write it up the best you can and post it on Arxiv. But if you haven't got a complete mathematical argument, what exactly have you got that makes you think you DO have a complete mathematical argument? Just playing Devil's advocate.

You can submit to Arxiv I think ... https://en.wikipedia.org/wiki/ArXiv

 

That would establish your priority.

 

There's also something called Vixra (Arxiv backwards) https://en.wikipedia.org/wiki/ViXra. The sense I get is that there's a lot more "fringe" stuff on Vixra. Either way those are two places to upload papers.

 

As far as whether you actually solved the problem, without seeing the paper I'd just want to know if you're familiar with all the modern literature on the problem, if you've found an original way to surmount some known obstacle, or if you have perhaps figured out how to extend or combine known techniques, all those things would add to the probability that you actually have a proof.

 

Your odds go down if this is just something you cooked up yourself. Someone's probably thought of it already and if they couldn't get it to work, that would be a problem.

 

I'm not an attorney but as far as I know you could publish your proof right here. If it turns out to be correct and your proof becomes famous, your IP address assigned to you by your Internet service provider, combined with the Scienceforum server logs, would establish your authorship under your real name.

Thank you very much for your detailed attention to my post. I hesitate to answer before I am at least able understand the points you are making and how they apply to what I was trying to say.

If anything I wrote is unclear I hope you'll give me the chance to make it clear.

 

I encourage you to ask all the questions you have. Please don't burden yourself with having to figure out basic topology before you feel worthy of asking a question.

 

I will have to give it a bit of time and then perhaps I will be able to answer.

 

I will only answer,though if I feel that I have gained a preliminary understanding of the points you have raised as it is not helpful to anyone for me to add confusion on top of my initial confusion.

Confusion is normal. Better to ask questions.

 

 

You did ask me a specific question or two but I still prefer to also leave that till later (if I feel that I can formulate a decent response)

You can ignore what I said about not understanding your torus construction. Strange pointed out that you are thinking of solid balls and I'm thinking of hollow spheres. By default when you say "sphere" in math it means a hollow one. So n-spheres always live in n+1 dimensional space. The 2-D sphere lives in 3-space. That's the official terminology for what it's worth. If it's solid, it's a ball. Also note that the standard torus is hollow as well. There's no special name for a solid torus.

 

Now that I understand that you are thinking of solid balls I agree with your construction. You are drilling a hole from one side of the earth to the other, and what's left is a torus. In fact you are left with a solid torus.

 

I must say I have no idea whether this idea generalizes to higher dimensions. Geometric visualization was never my strong point and your guess is as good as mine. I think of (hollow) torii as Cartesian products of circles. I find that easier to visualize than your idea of drilling through solid balls. But I'm sure the two approaches are equivalent.

Anyway ,one of the examples that always comes up is the torus and I have noticed that you can make a torus by inserting a space in a 3d sphere or globe and then stretching the "remainder" of the sphere to that of a torus or indeed to any object that shares the same topology as the torus.

Your instincts are good. I think you'd benefit from picking up some mathematical concepts and terminology, and tightening up your exposition. I have some remarks along those lines.

 

* First, note that the sphere and the torus are both two-dimensional manifolds. That means that at each point, if you just look around a small area, it's "almost" a plane. In the same sense, a circle is a 1-dimensional manifold. You might take a look at that link. It's a key idea in the mathematics of studying shapes.

 

Also by the way, dimension is a very complicated topic, far more subtle than it appears. For example shapes can have fractional and even irrational dimension.

 

You are correct of course that a sphere is embedded in 3-space, and a circle is embedded in 2-space. That's the technical term. But we are careful to distinguish between a shape itself, like a sphere, and the space it's embedded in. There are in fact some 2-dimensional shapes that can't even be embedded in 3-space, for example the Klein bottle.

 

* Secondly, I I don't follow your construction. If you take a sphere and poke a hole in it, or remove just a single point, you can flatten out what's left to a circle or square in the plane; and in fact you can stretch it all the way out to be the entire plane. So in fact your idea to poke a hole in a sphere leaves you with the plane, topologically.

 

To make a torus out of a sphere you have to poke two holes, then glue the circumferences of the holes to each other. Is that what you are trying to say? It would be helpful to clarify this idea. Someone mentioned mathematical surgery and that's what this is.

 

Finally, without going into detail, the best way to describe a torus mathematically is as the Cartesian product of two circles. https://en.wikipedia.org/wiki/Torus

 

The "method" I am suggesting is to start with a ,,one dimensional (I was thinking of a 3-dimensional .object but clearly if I can't start with 1 dimension then there is no point)

 

This one dimensional object (a straight line) is a set of points with perhaps 2 extremities .

Your idea of extremities is unclear. Let me give you some examples.

 

In topology there's a notion of boundary points. For example we'd all agree that the open unit interval [0,1], the endpoints 0 and 1 are boundary points.

 

But what about the open unit interval (0,1)? By definition that set does not include its endpoints. But in topology we still consider 0 and 1 to be boundary points.

 

Now there's a problem for your idea, because the open unit interval (0,1) is topologically equivalent to the entire real line; which clearly does not have any boundary points. Having boundary points is not a topological property.

 

With this in mind, can you clarify what you mean by extremities?

 

 

Next we introduce an "exclusion zone " ( a gap) which forces the other points in the set to configure themselves around it in a continuous,organized way -a bit like the way the surface of running water deviates around a rock projecting from the water..They can stretch to any shape provided there is this gap .And the "gap" can increase to any size and shape also.

I confess I can't follow this at all. You just seem to be poking a hole in a surface or making a tear in it, but the concept of an exclusion zone is unclear. And in topology you can always stretch things as much as you want, so that "size and shape" are not relevant. A big square and a small circle are exactly the same topologically.

 

 

Once this new configuration is made it may be possible to introduce a second gap in the same way and that would give a new , more complex one dimensional topology.

No doubt you can put holes in things to change their topology. But this doesn't seem too significant. Rather it seems obvious. If I poke a hole in a sphere I no longer have a sphere.

 

I think your visualization is clear to you but not to me.

 

This process would work in exactly the same way for any-dimensional object and the "gap" or "exclusion zone" would be correspondingly dimensional giving any amount of n-dimensional topologies.

Well this is unclear to me in any dimension. Yes it's true that if you have a topological object and you put a hole in it you'll get some different topological object. But that's practically by definition, since topology is the study of properties that are invariant under stretching and moving but not tearing or putting holes in. So if you put a hole in a manifold you get a different manifold.

 

 

Now I realize this may be seriously (and perhaps laughably ) flawed but that is what I had in mind

I don't think it's flawed but it's vague. I think your instincts are good and you'd benefit from reading about manifolds and general topology; and also spending some time trying to tighten up your ideas.

It is both a cubic and a quartic. Polynomials are best understood as "nesting" - all constants are linear, all linear polynomials are quadratic, all quadratic polynomials are cubic, etc.

No polynomial with nonzero A, correct.

I understand your point about nesting, but I don't think your terminology is standard.

 

The degree of a polynomial is the largest exponent of any term with nonzero coefficient. [math]x^3[/math] is a cubic and not a quartic. Nothing is gained by obfuscating this simple fact.

 

As evidence for my point of view I would simply submit the common usage in algebra of the degree of a polynomial. If a cubic is a quartic is a quintic, then the definition makes no sense. A polynomial of degree 3 is perfectly well defined, and it's called a cubic. There is no ambiguity in this terminology.

OK you are nearly there.

 

Imagine that you have a small rectangular patch, of width dx and length dy, standing on the xy plane and have erected a rectangular prism to just touch the underside of some function z = f(x,y) as shown in the sketch.

See, this is exactly the kind of question I would never touch And I won't even try to pin you down by asking you exactly what width dx means!!

Written whilst you were still editing.

Just getting the quoting right takes all I've got!

 

Thank you for post 29, it is most interesting and illuminating, so I hope you won't take my comments the wrong way.

Not at all, I got some of my own thinking straight too. In particular, I realized that without AxInf, the natural numbers form a proper class in ZF-minus-AxInf. However that still would not satisfy the ultrafinitists.

 

I have been discussing this topic with a couple of pure math professors elsewhere, who very definitely use real and potential infinities, however see my SCIAM link.

That link is about extremely advanced set theory and has no relevance to our conversation. The state-of-the-art thinking about the Continuum hypothesis is far beyond anything we're talking about.

 

 

 

I have heard (and the SCIAM link attributes) your definition (2) attributed to Cantor, not Dedekind, though I understand Cantor was good at acknowledging the work of others, unlike some.

 

Here's the Wiki page for Dedekind-infinite_sets.

Whether this is a case of the wrong person getting credit I cannot say. The Wiki page says Dedekind came up with the idea and/or name in 1888, so I imagine he and Cantor must have exchanged ideas.

 

 

As far as I can see, you have not proven that ZFC minus the axiom of infinity disbars the existence of an infinite set, merely that it does not provide one.

You're correct. I owe you that proof that if there's no inductive infinite set (which is what AxInf gives) then there's no infinite set at all. But the natural numbers (as I defined them earlier) are an inductive set; and the negation of AxInf says there is no inductive set. So ~AxInf definitely says that there is no set that models the natural numbers. Let me think about the full proof that there can be no infinite set at all, not even a non-inductive one.

 

 

I did wonder if your statment referred to the 2010 Cohen proof?

Paul Cohen died in 2007 though his spirit lives on. Not clear what you're referring to. In any event none of this is related to Cohen's famous proof of the independence of AC and CH, which uses advanced set theory and is not related to our much more elementary concerns.

 

 

 

 

 

 

 

 

 

 

 

 

 

http://www.scientificamerican.com/article/infinity-logic-law/

That article refers to the absolute bleeding edge of modern set theory, and is not relevant to our concerns. We are in ZF and at most ZFC. The theories they're talking about involve the addition of many so-called large cardinals, which go far (far far far ...) beyond any sets we are contemplating. Large cardinals are sets that are too big to live in ZFC and require new axioms for their existence. What that article is about is the question of how many and what type of large cardinals can you add to ZF without making set theory inconsistent, yet resolving the Continuum hypothesis. Woodin has an idea but his proof's not even finished as far as I know.

 

An analogy would be that we are discussing the fact that molecules are made of atoms and atoms are made of protons, neutrons, and electrons. And the SciAm article is about the most advanced as-yet-unproven work in string theory. There's a gap of a century of professional research between atoms and strings, and between elementary ZF and the type of set theory that article is discussing.

 

 

 

(Added later ...)

 

Ok, about that proof I owe you ... it's trickier than it looks.

 

The question is whether the negation of AxInf, which says there is no infinite inductive set, implies that there is no infinite set.

 

First, proof by Wiki. From https://en.wikipedia.org/wiki/Infinite_set:

 

The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in ZermeloFraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers.

 

Of course that's not actually a proof, it's a claim without a proof. And it's interesting that they explicitly use ZFC (ZF plus the axiom of choice) and not just ZF. Could it be that this proof requires choice?

 

 

I found this lovely little proof here. It's Henning Makholm's comment to the original question.

 

Theorem: There is no infinite set in ZFC-AxInf.

 

Pf: If X is any set in ZFC-AxInf, by the axiom of choice we can well-order X. That means that X is order-isomorphic to some ordinal; and in particular, X is bijective to some ordinal.

 

Since ~AxInf says that there is no infinite inductive set, it follows that there are no infinite ordinals. (Inductive sets are ordinals). Therefore X must be bijectable to some finite ordinal, ie a natural number.

 

QED

 

In order to drill that proof down to include a detailed explanation of ordinals and inductive sets, the post length would quickly get out of hand. I started doing that then realized it would just be another way-too-long post. Perhaps it's better if I leave it as-is and respond to specific questions.

 

To sum up: There are no infinite sets in ZFC-AxInf.

 

Now what about ZF-AxInf? That's ZF minus AxInf and also without the axiom of choice. Well, I can't find a proof and I can't think of one!

 

I wonder if perhaps it's not even true. When you take infinite sets without choice, you get weird consequences, such as an infinite set that's not Dedekind-infinite. Perhaps there's some pathalogical counterexample in ZF-AxInf of an infinite set. I will continue to look around.

Can you outline or link to the proof please?

https://en.wikipedia.org/wiki/Hereditarily_finite_set

 

Also: http://plato.stanford.edu/entries/set-theory/, from which I quote:

 

The theory of the hereditarily-finite sets, namely those finite sets whose elements are also finite sets, the elements of which are also finite, and so on, is formally equivalent to arithmetic.

 

Also http://www.isa-afp.org/entries/HereditarilyFinite.shtml, from which I quote:

 

An HF set is a finite collection of other HF sets; they enjoy an induction principle and satisfy all the axioms of ZF set theory apart from the axiom of infinity, which is negated. All constructions that are possible in ZF set theory (Cartesian products, disjoint sums, natural numbers, functions) without using infinite sets are possible here.

 

 

 

 

 

 

 

 

 

Since you want me to define infinity, I take it you are prepared to define an infinite set?

There are two common definitions of an infinite set.

 

* (Infinite) A set is called finite if it's bijectable to some natural number. Otherwise it's called infinite.

 

* (Dedekind infinite) A set is Dedekind-infinite if it's bijectable to a proper subset of itself.

 

https://en.wikipedia.org/wiki/Dedekind-infinite_set

 

There are some subtleties and considerations I'd like to mention.

 

* What do I mean "bijectable to a natural number?" In what way is the set {Socrates, Plato} bijectable to the number 2? Good question! The answer is that in set theory, the number 0 is defined as the empty set; the number 1 is defined as the set containing 0; the number 2 is defined as the set containing 0 and 1, and so forth. So in fact 2 = {0,1}, a set that conveniently has exactly 2 elements. This definition by the way is due to John von Neumann.

 

Using this definition, we can implement a model of the Peano axioms within ZF, with or without AxInf. Without AxInf we have 1, 2, 3, ... but not a set containing them all. AxInf allows us to write {1,2,3,...}, a set that contains all the natural numbers. As the references I linked indicate, we don't need an infinite set to do arithmetic and mathematical induction.

 

* If we assume the axiom of choice (AC) then a set is infinite if and only if it's Dedekind-infinite. But strangely, if we assume the negation of AC, these definitions are not equivalent. That is, in the absence of AC, there is a set that is not bijectable to any natural number; yet still has no bijection to a proper subset of itself.

 

[in passing I note that people sometimes dislike AC because it leads to counterintuitive results. But the negation of AC also leads to counterintuitive results. Pick your poison. We generally assume AC because it's convenient to do so.]

 

* Finally, a point I made earlier. A bijection is defined in set theory as a type of relation that maps a set to a set. Without an infinite set of natural numbers, you don't have a bijection from the natural numbers to a subset of themselves. In the absence of AxInf there is no infinite set and no Dedekind-infinite set.

 

Now, here I think is the nub of the matter.

 

Consider the collection (not set) 1, 2, 3, 4, 5, ... I think we agree intuitively that is an infinite collection, even in the absence of AxInf. It may not be an infinite set, but it's an infinite something.

 

In set theory we have a name for a collection that's "too big" to be a set: a proper class. So in ZF-minus-AxInf, the natural numbers form a proper class. In standard ZF, which includes AxInf, the natural numbers form a set. A more familiar proper class is the class of all sets. It's too big to be a set by Russell's paradox.

 

Aristotle made the distinction between potential and actual infinity. I believe that this is the exact distinction between an infinite collection and an infinite set. The sequence 1, 2, 3, ... is a potential infinity. The set {1, 2, 3, ...} is an actual infinity. This terminology is NOT used in math, only in philosophy; but perhaps this is the point you are making.

 

Hope this is helpful. If you are saying that even in the absence of AxInf, if we believe in the natural numbers then there is an intuitively infinite proper class, then I agree with you. But do note that the ultrafinitists don't even grant that much! They don't believe in the full collection of natural numbers.

d.

Ok, then what is your position exactly? You said I don't understand what you're saying -- and I agree. But when I ask you directly to state your thesis, you don't. That's why I don't understand your thesis. You have not clearly stated it.

 

 

 

Take the collection of set axioms of your choice (pun intended for your pun collection)

Remove the axiom of infinity.

For purposes of this question, I'll start with standard ZF and remove AxInf. So I'm working in ZF minus AxInf.

 

Now my question is

 

Is the reduced collection of axioms

 

a) Compatible with the existence of infinity

No, in ZF minus AxInf, the nonexistence of an infinite set is provable.

 

But I do want to make one refinement. You say "infinity" but that's imprecise. I am using the phrase, "an infinite set," and that is precise. Under the negation of AxInf, an infinite set does not exist, provably.

 

So please define what you mean by "infinity." Do you mean an infinite set? Or do you mean something else, and if so, what?

 

b) Incompatible with the existence of infinity

The negation of AxInf is incompatible with the existence of an infinite set.

 

Since you haven't defined what you mean by "infinity," I can't answer.

 

 

Regarding your question about Mr. Jines, I take it you are referring to factual assertions about fictional entities. For example, Ahab is captain of the Pequod. True. Ahab is the cabin boy. False. So we can make true/false claims about fictional entities. But please, let's focus on what you mean by "infinity," because I'm starting to suspect that when you say "infinity" you mean something OTHER than an infinite set; and that would indeed cause confusion until you define what you mean.

That is the problem.

 

You appear to have reverted to

quoting tracts of what I say,

Declaring it false.

But not actually addressing the content.

You might have posted that before my last edit. You are right, I don't understand what you're saying. Which of these is closer to your claim:

 

a) An infinite set exists if and only if we assume AxInf [that's my position]; or

 

b) An infinite set exists whether or not we assume AxInf. If we don't assume AxInf then we can't PROVE an infinite set exists, but in the Platonic realm an infinite set does in fact exist [that's not my position, because I'm not a Platonist]; or

 

c) Whether or not we assume AxInf, an infinite set exists, not just in the Platonic world but ... well, where exactly?

 

d) Something else.

I think we are still at cross purposes.

To rectify that situation I will state my purpose.

 

I am explaining. I assume you are interested in learning.

 

Why would I say that? Because in other posts you have said that your background in math is practical and applied. Mine is pure and abstract.

 

If you are talking to me about set theory, I must assume you have curiousity about the subject. I'm the right person to explain it.

 

Of course if we were discussing applied math, the situation would be reversed. I would not presume to argue with you about Fourier series, PDEs, or even integration by parts. When I taught calculus most of my students could do integration problems better than I could. That's only a slight exaggeration.

 

But when it comes to basic set theory, I'm authoritative.

 

That's where I'm coming from, for better or worse. I assume if you're talking to me at all, it's with the intention of trying to learn how pure math sees the subject of infinity. What other reason could there be?

 

So ... where are you coming from?

 

 

 

The only meaning of "there exists an X such that" in mathematics is, as I have already indicated, that the stated properties of X are consistent with the definitions, axioms and deduced theorems we are currently employing.

That's right. If you assume the axiom of infinity (AxInf from now on) then there is an infinite set. If you assume the negation of AxInf, there is not an infinite set. Note that there may still be an unending sequence of natural numbers 1, 2, 3, ... But there is no (completed) set of them.

 

I use the word completed because that was Aristotle's clever idea, to distinguish between the potential and the actual infinity. However note that the terms potential/actual do not occur in modern math, only in the philosophy of math.

 

[aside] Don't forget it has been shown that there is no comprehensive list of definitions, axioms and theorems for the whole of mathematics at once. The progroms of Hilbert and Frege are unrealisble dreams.[/aside]

Non sequitur. Totally irrelevant to our discussion. "Argument from random Wiki pages without understanding."

 

It does not matter whether we have 'discovered' X or just invented it or whether X is an abstract construct in our minds or has physical manifestation, mathematically X 'exists'.

I could invent pink elephants with green spots, for the purpose of counting spots on an animal. Mathematically, such an animal would exist.

This is really no different from saying Mr Jines' garden is 50' by 30' in a problem in a maths book.

There is no necessity for Mr Jines to physically exist or have a garden if he does.

Right. Assume AxInf and an infinite set exists. Deny AxInf and no infinite set exists. I am puzzled as to why you keep ignoring this point.

 

To be more accurate: Assume AxInf and the existence of an infinite set is provable. Deny AxInf and the existence of an infinite set is not provable. The "truth" of the matter is unknown, nor is it clear that there is a truth of the matter at all.

 

 

Therefore I assert that at least one infinite set exists, and therefore at least one infinite 'object', since object is a more general noun that includes set.

I do not know of any inconsistency with the current definitions, axioms and theorems.

Yes you keep ASSERTING. What you have not yet done, and what I have repeatedly invited you do to, is to put forth an actual ARGUMENT in support of your position.

 

What "current definitions, axioms and theorems" do you refer to?

 

In standard ZF, an infinite set exists. In ZF minus AxInf, no infinite set exists. Why do you persist in ignoring this point?

 

 

 

 

If you can offer one or more, please post them.

I have repeatedly stated the argument. The system ZF minus AxInf is a consistent set theory known as the hereditarily finite sets. It has W Wikipedia entry that I've already posted.

 

Tell me: What do the hereditarily finite sets mean to you? Do you understand their significance? Please read this page. https://en.wikipedia.org/wiki/Hereditarily_finite_set

 

[aside] Sometimes we find that our rules conflict with experience. As such we need to re-examine our rules. Such was the case with the introductuion of i, which I hope you will grant mathematical existence to. [/aside]

More irrelevance. I don't see an argument here. In fact in math when the rules are logically consistent yet conflict with our experience, it's the rules that matter. Such was the case with non-Euclidean geometry (an example you brought up earlier). Such is the case with the mathematics of the infinite.

 

Please tell me what your point is. We actually have made no progress at all since our previous posts. You still have not put forth an argument that an infinite set exists even in the absence of AxInf. Is that in fact your claim? I can't figure out what you are claiming.

 

ps -- Perhaps to clarify in my mind exactly what it is you are saying, which of these are you saying:

 

a) An infinite set exists if and only if we assume AxInf; or

 

b) An infinite set exists whether or not we assume AxInf. If we don't assume AxInf then we can't PROVE an infinite set exists, but in the Platonic realm an infinite set does in fact exist; or

 

c) Something else.

No, I've never come across that one.

Thank you.

You're welcome! I'm addicted to bad puns and lame jokes, I got a million of 'em.

 

 

 

Yes I couldn't agree more, I've even started a thread here asking for examples of words which cause confusion because of multiple meanings in different disciplines.

Thank you for the calculus example, I will add it to my list.

Set! That's causing the confusion right here!! Set has a very technical meaning, it couldn't be more different than the everyday idea of a collection of things. Likewise group, ring, field, module, manifold (the thing on your car's engine) etc. I can't think of any word in math that does have its usual meaning. I didn't see your other thread, if you link to it I'll be happy to add to it.

 

I agree that truth is one such word, and therfore perhaps best avoided.

I hope it's not me you're agreeing with, since I never said that the word truth should be avoided. Truth has a very specific meaning in mathematical logic and it's been intensively studied. And it's the subject of our conversation: Whether it's true that infinite sets exist in math, or just a convenient assumption. I'm arguging the latter. I believe you're claiming the former but not putting forth an argument.

 

Come now I think "just a convieniece is a bit weak", don't you?

I would say it's a strong statement. I am making a very specific point. Infinite sets are widely accepted in math because they're useful and interesting. Any claim of truth beyond that requires careful analysis and exposition. I'm sure one could make a case that infinite sets "truly" exist in math; just as well as one could make the opposite claim. We accept infinite sets because they're useful. This is a fact. It's counterintuitive because the axiom of infinity is always assumed in ZF[C], so it becomes like the air. We don't notice it unless we are explicitly discussing axiomatic set theory. But isn't that what we're discussing here? If not, maybe that's a source of confusion.

 

To put this in a contemporary context, intuitionism and constructivism are making a comeback, due to the influence of computer science and also category theory. (That's a big statement which would take us too far afield to go into right now in any detail). There are a lot of people these days who feel that a mathematical object exists only insofar as it can be explicitly constructed by an algorithm. So pi and the square root of 3 exist; but not the uncomputable numbers. There are a lot of people doing serious work in this area and it's far from clear what our ideas of the real numbers will look like in the future.

 

If one believes in full powersets, one goes immediately from the naturals to the reals. But most subsets of the reals can not be generated by an algorithm and have a far weaker claim to existence than, say, the number 6 or the set of even numbers.

 

By the way I myself am not a constructivist. I believe in full ZFC. That doesn't mean I think it's true. It means I think it's convenient! But I'm aware that the constructivists are making a lot of progress lately and it's far from clear which point of view will prevail. Math is a historically contingent human activity, whether or not there's an ultimate truth "out there."

Well what of Euclid's 5th axiom then?

It's logical to accept it and logical to reject it, but we don't have all this ho-hah about Euclidian v non Euclidian geometry.

Both are equally accepted into mathematics as consistent.

What a perfect example supporting my point. When the young Gauss discovered the consistency of non-Euclidean geometry in the 1820's he did not publish, fearing the negative impact on his career. Imagine that, Gauss himself afraid to publish. That gives us an idea of just how controversial non-Euclidean geometry was. When Riemann and others established non-Euclidean geometry in the 1840's it was VERY controversial and shocking to people. It was regarded as a mathematical curiosity having nothing to do with reality. It wasn't till Einstein came along with his crazy theory of relativity, and Minkowski said, "Hey, math has just the right gadget in our back pocket." That's when people started taking non-Euclidean geometry seriously.

 

There was a MASSIVE hoo-hah about non-Euclidean geometry before that. Simply massive. It was really upsetting to people. Think about it. One day, math and physics are the same thing. The next day, they're not. This was one of the most profound intellectual developments in history. People are STILL arguing about the extent to which math must be based on reality.

 

 

For the rest I think you are manufacturing an argument, where none exists.

I'm making an important point of mathematical ontology that you're failing to understand or engage with. If you claim that infinite sets exist in mathematics independent of the axiom of infinity, make a case. Or if you think that they do depend on the axiom of infinity but that the axiom of infinity is somehow more correct, more natural, more true, then make that case. I claim it's simply more convenient. If you disagree, don't just repeat your claim. Make an argument. Maybe I'll learn something.

 

 

I made it perfectly plain that the mathematical statement

 

"There exists an n such that n+2 = 3" doesn't give physical embodiment to the phrase "there exists", as does English,

it actually means that n = 1 is consistent with the rules of arithmetic.

I'm not entirely sure what this has to do with the discussion. I also don't recall you making this statement. This is a two month old thread that someone resurrected and I haven't read through from the beginning, would that help?

 

 

 

 

By the same token the mathematical statement

 

There exists an infinite object means that we can demonstrate a mathematical object which can be placed into one-to-one correspondence with a part of itself, that is not inconsistent with stated mathematical rules, though it may contravene others that we are not employing. It does not mean we can, as I said, buy a pound of it in Tescos.

An infinite object? Oh yes. An infinite SET? Requires the axiom of infinity. Did you notice that you swapped in the word "object" when we are talking about sets? The word set has a lot of technical baggage around it these days. One can agree that there is an infinite object without agreeing that there is an infinite set.

 

Let me sketch a proof that in the absence of the axiom of infinity, there is no such bijection. I'll grant you the existence of a collection of natural numbers 1, 2, 3, 4, 5, ... Now you say there's a bijection between ... well, between what, exactly? A bijection is a type of function. What is a function? It's a mapping that goes from a set to a set. That's the formal definition of a function. So now that we haven't got a set of natural numbers, we can't biject anything to it. You see how tricky this can be. Saying something is a collection is fine. Saying it's a set means that we can define functions on it, take powersets, etc. Without the axiom of infinity we can't do any of those things to the collection of natural numbers.

 

I hope this is clear, it's the heart of the matter. The axioms tell us exactly what operations may be performed on sets. If we have a collection that's not a set, we can't do those operations on it. You haven't got a bijection between the natural numbers and anything, because without the axiom of infinity, you haven't got a set on which to define a function. You may well have an intuition about these things, so do I. But if we are being careful to follow the rules of set theory, we have no bijection.

 

 

Yes I assert that mathematical infinity exists

You can not possibly justify that statement without adding context. Are you saying an infinite SET exists? If so, that requires the axiom of infinity; and it's perfectly consistent to deny the existence of infinite sets; and the only reason we accept infinite sets in everyday math is that they are convenient. There is no other reason. If you have an argument to the contrary, you are free to make it but so far you have not done so.

 

, but on the other hand remember that mathematical infinity has a different meaning than the one you are perhaps used to in English.

I wonder what I could ever have written that leads you to think I'm confused on that point?

Of course it does.

I envy people who have such certainty. Certainty is the fashion these days. You can't be thoughtful on cable tv. Opinions must be loud, certain, and extreme. But when we're discussing a philosophically loaded aspect of math such as infinity, may we dispense with absolutism and perhaps acknowledge the difficulty of the question? Or are we on cable tv here too?

 

If we accept the axiom of infinity, then an infinite set exists.

 

If we reject the axiom of infinity, then no infinite set exists. Note that we still have all the natural numbers 1, 2, 3, 4, 5, ... What we don't have is a set that contains all of them. This is a perfectly valid system, known as the hereditarily finite sets.

 

Ever since Russell's paradox, we learned that we can only apply the word "set" to very specific collections that satisfy some formal axioms. We can't just put any old collection of objects into a set just because we want to. We can't just say, "Well I can imagine 1, 2, 3, ..." therefore there's an infinite set. In fact that's not a valid logical deduction. All we have is an infinite collection that may or may not be a set.

 

If it's logically consistent to accept the axiom of infinity and logically consistent to reject it, why do we generally accept it? Simply for pragmatic reasons of convenience. It's a lot easier to do math if we allow infinite sets, so we allow them. This has of course no bearing whatever on the truth of the matter, if it's even meaningful to speak of truth in this regard.

 

Accepting infinity into mathematics is simply a convenience, nothing more.

 

One can go farther. There are ultrafinitists who reject not only infinite sets; but even sufficiently large finite sets. The late Ed Nelson was an ultrafinitist who was a serious, well-respected mathematician.

 

In summary, there is no absolute answer to the question of whether mathematical infinity exists. It's purely a matter of personal choice and convenience.

 

Or did you mean perhaps that Cantor's work shows that infinite sets exist? Sadly not. There's a lot Cantor didn't know about set theory, just as there's a lot the Wright brothers didn't know about airplanes. Set theory in its modern form took a good fifty years to develop after Cantor's work in the 1870's. What Cantor thought of as sets wouldn't pass muster in undergrad set theory class today. That doesn't diminish his genius, but it does indicate the logic traps we may fall into if we take Cantor as the last word about set theory rather than the first.

 

But equally of course

Sorry, couldn't parse that. What does that refer to?

 

 

The mathematical statement "there exists" does not mean you can go down to the supermarket and buy a pound of "infinity".

It means that the properties of a mathematically defined object called "infinity" is consistent with the axioms and theorems already available.

Which axioms? You said "of course" to the question of whether mathematical infinity exists. But now you admit that it's purely a matter of which axioms one adopts and that there is no actual truth of the matter (or that the truth of the matter is not presently known). Which is your position? You are not being clear. You appear to be contradicting yourself in two different paragraphs.

 

However we should be aware that the words cardinal and ordinal have different usage in English and mathematics.

Well yes, just as the word vector has a technical meaning in math that's subtly different than the meaning in physics which is a lot different than the meaning in biology. Just as calculus is the stuff the dentist scrapes off your teeth. We're having a math discussion. You know the old joke. Q: What do you get when you cross a mosquito with a rock climber? A: Nothing. You can't cross a vector with a scalar.

... there is a number system that can prove the cardinality of the different levels of infinity(olafs) ...

The Flying Olafs were a group of Norwegian acrobats in the 1930's. Perhaps you're thinking of the set theorist who walks into a bar and says, "Give me a martini please, and hold the Aleph."

Hmm... Seems little point to continue on, but I will, because why not.

 

Any one could do this if they tried, although I am pretty good at mental calculation and concentration.

I notice you don't seem to write down your carries. Do you keep them in your head as you're computing the sum of the next column? Or do you write them down on a different piece of paper?

 

This is very impressive work. I am even more interested in the teacher-student dynamic here. They clearly recognized that you have a special aptitude for this kind of work. A few hundred years ago mathematicians and physicists had to do this all the time. I imagine Newton doing something similar when he worked out that the motion of the moon fit his theory of gravity.

 

This particular kind of skill, organization, and focus at hand calculation is very rare these days.

Impressive! Keep going!

 

Are you really good at this sort of thing in general? Do you have strong mental calculation skills and concentration?