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That's exactly what it means, by definition. Any set with the same cardinality as the naturals is countable. If it's strictly smaller it's finite. If it's strictly larger it's uncountable. That's all the word uncountable means.

I guess this characterization of what is alleged by Ford to be "an immature moment" is what really bothers me. Women will never be safe if men characterize this type of behavior as an 'immature moment' instead of a serious assault. This was not a case of a boy trying for second base when she only wanted to go to first base. Regardless of whether or not the assault occurred, to dismiss it so easily does a disservice not only to women but to all of us.

Mine is to trust in God and to be good. I know it sounds childish "be good" part but that just means try being good person by following God's and Jesus Christ commands.

There is a disgraceful post containing full on gay pornography hanging for hours in the lounge section. I’ve reported the post few hours ago but apparently there arent any mods online to take it off. There are quite a few regulars online though, could we give an ability to hide posts to regulars like @Strange or @studiot or @iNowif they agree to it? There are quite a lot of hits to this site per hour and a hanging pornographic photo along posts containing extreme violence is not something we want this forum to be related to. Maybe we could do a vote who we would want to have an ability to hide posts at times when no mods are online? @hypervalent_iodine, @swansont, what do you think? Edit: I am not shooting to be one of the people who will have the ability to hide posts, I am in for a few months of intense work and I would not be of use.

The 'laws' of the Physics ( IE the universe ), are inherent in the differentiable symmetries of the action of a physical system. Look up Noether's theorem.

OK. Lets take gravity as an example.Our best theory of gravity is described by the Einstein field equations: [math]R_{\mu \nu} - \tfrac{1}{2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G }{c^4} T_{\mu \nu}[/math] So when you say "where is that", what exactly do you mean? It is there on the page in front of you. You can find it in text books. Some people hold it in their heads. But you won't find it written in space. So what are you asking? BTW if you want a good overview of what the equations mean, you can find it here: http://math.ucr.edu/home/baez/einstein/ There is some math, but you can understand a lot even if you just skip that.

Why throw a god into the mix? But it may be OK for a thought experiment.This will me very imprecise; the purpose of my comment is to express an opinion in context of the thought experiment, not present any scientific facts. Since I am a practical guy relying on mainstream science and observations, I would: 1: remove space 2: create a hot dense state and let it evolve as described by the Big Bang model. Ok. What scientific experiment do you suggest to tell if one of us is right? Is it possible within the scope of science to test it? Can you provide some evidence for your pixel based universe other than "it is self evident"?

Yes sir. Quite right! In this instance, though, that will be happening on Thursday. Again, I have no reason to disbelieve her, but I acknowledge others do. That’s okay. What’s interesting is that most of us would like to see this go to court. It won’t because this is about politics and winning not justice and truth. He’ll be on the court for 40 years. Surely we can spend 40 days making sure it’s the right thing to do?

electricalelectronics has been spam-banned for posting vulgar and pornographic material. Apparently he either had a meltdown after being warned about some marginal posts, or was planning on being a jerk all along. Spam-banning removes all of his posts. Apologies if this disrupts any ongoing discussion, from before he decided to react like a child.

Your opinion is that the universe is pixellated or cellular. This is an opinion because there is no evidence for it. Note that the idea has been suggested many times (so this is not some great insight you have come up with). And then people have come up with mathematical models based on the idea (something you have failed to do). They have then used these models to make testable (quantitative) predictions (again something you cannot do). Experiments have then been done to test these predictions. No evidence for the pixellated nature of space has been found so far. This process is called "science". Asserting that your idea must be right because it seems obvious to you is not called science. This is a science forum. Do you see where the problem is? By your definition of "objective fact" as in the table example, we should all be able to see and agree that the universe is pixellated, but in fact we can't. So it is not a fact. As you say there are only two choices, it must therefore be opinion. And can you use this to calculate the orbital height of a geostationary satellite? That would allow us to check the accuracy of your theory. Then you are on your way to getting this thread closed and maybe banned from the forum. That seems a very silly attitude.

So, suddenly, a very specific request has been answered by a more general response. That's what I'd expect from wishful thinking. Next time, get Him to send an alligator. That won't look so much like a meaningless coincidence.

I see two problems here already. 1) The diagonal argument is NOT Cantor's theorem. There is no "list" in Cantor's theorem. Cantor's theorem says that any set has strictly smaller cardinality than its powerset. https://en.wikipedia.org/wiki/Cantor's_theorem. This is why you need to be more clear. 2) Neither Cantor's theorem nor his diagonal argument need to be framed as proofs by contradiction. The diagonal argument is a direct proof that any arbitrary list of reals is missing at least one real. Cantor's theorem is a direct proof that an arbitrary map from a set to its powerset fails to hit at least one set. Any complaints about proof by contradiction are misplaced here. > Does the power set of ℕ exist at all? Yes, by the axiom of powersets. https://en.wikipedia.org/wiki/Axiom_of_power_set > If it exists, its cardinality would be aleph 1 No, that depends on the Continuum hypothesis and is independent of the other axioms. The best we can say is that the powerset of [math]\mathbb N[/math] is [math]2^{\aleph_0}[/math]. But nobody knows which Aleph that is.

! Moderator Note This style of ignoring what people say about your assertions so you can make additional assertions without supportive evidence isn't conducive to discussion. If you can't take the comments on board, perhaps you should start a blog on a different site. Start engaging with substance and rigor or this thread will be closed.

Important to understand the difference between potential and actual infinity: https://en.m.wikipedia.org/wiki/Actual_infinity The program only potentially passes through infinite number of states it never actually achieves Infinity...

In the Solvay Congress of 1927 the use of the word probability as the name of a new mathematical procedure, typical of quantum theory, different from the procedures that the same word names in thermodynamics, has been accepted. I want to start confessing my perplexity and my disagreement in front of intruducing the postulate that proposes to express the square of the wave amplitude in terms of probability. That requires admitting that the square root of probability is something that has the physical ability to propagate. I find this idea as deficient as saying that the number concept of mathematics occupies space and is capable of traveling. If DannyTR feels the same, then in that I agree with him. In other details I do not agree and I will try to show why. First let's think about the natural numbers. And suppose there is no infinite. Then, in increasing sense, there is an insurmountable maximum [math]N_f[/math] in the set of natural numbers. You generate a recurring program, which adds 1 to the result obtained in the previous step. In the first step there is no previous step, then the previous result is zero. Sum 1 and the result is 1. The second step is to add 1 to the result of the previous step, that is 1 + 1 = 2. With this recurrent procedure, the program passes in increasing order by all natural numbers. In case there is no infinity in natural numbers, the program should stop when it reaches [math]N_f[/math] . The computer is enabled to continue, the adding procedure is enabled to advance without logical failure, there are no mathematical obstacles, but the program stops at [math]N_f[/math] . This is very weird. No multiplication, no potentiation, no operation with natural numbers could give a result greater than [math]N_f[/math] . The number of bosons in a physical system is natural. Physical laws explicitly state that there is no upper limit to the number of bosons that can coexist within a finite volume. Are the investigations that led to the formulation of those laws wrong? Note that the freedom to accumulate bosons unlimitedly is given for finite volume. That is, we can not present a critique based on the absurdity of infinite volume, because a finite volume is enough to accumulate bosons without limit. Numerical infinity within a finite volume. Obviously, an infinite number of bosons, within a finite volume, implies an infinite energy density of positive sign. Cosmology is beginning to admit that the net energy of the universe is equal to zero, because the energy of positive sign is compensated with energy of the opposite sign. Nothing prohibits that type of compensation can be established within a finite volume. If that volume contains two sets of symmetrical bosons, both with an infinite number of bosons but with opposite signs in energy, the energy density would be equal to zero within that volume. The numerical infinity linked to the zero of the net energy. Newton, Leibniz, Cauchy, Weierstrass, Bolzano, Cantor, each in his time, have studied infinity in mathematics. I translate a paragraph referring to the work of Bolzano, taken from a document in Spanish, available at the following address. https://www.palermo.edu/ingenieria/pdf2014/14/CyT_14_18.pdf --------- Translation: In the publication Paradoxes of the infinite of 1840 he claims the existence of the current infinite using the idea of set and recognizes that the difference between finite and infinite sets is the possibility of being in correspondence with a part of his own, as Galileo had already observed. End of translation --------- Everything known points against us when we try to underestimate infinity. That does not mean that we can attribute to infinity properties that do not possess, or invoke infinity in cases that do not admit it. That is why I wanted to give the example of two infinite symmetric energy densities, within a finite volume, that give a net density equal to zero, without forbidding the presence of an infinite number of bosons.

But dear Albert was being facetious. The size of any set of all real numbers that lie between any pair of real numbers is infinite.

Infinity does exist mathematically; it is well defined. As for whether it exists in the real world, that depends whether the universe is infinite or not! Yep. We know that.

Two variations on the computation by Timo (nice to see you), which change the 45TWh figure hence the cost. Generating electricity in one country isn't necessary and does not correspond to present-day practice. If the wind doesn't blow in whole Germany, it does in Scotland, Brittany, Aquitaine or Galicia. Electricity is presently transported, typically on such distances, as the market is continent-wide. There is no need to store an amount of (Germany's mean consumption) 64GW over 29 days as the 45TWh imply. Even if the electricity came from Germany alone, wind wouldn't stop for that long. Europe-wide, you won't have more than 1 day without wind. That would be 1.5TWh storage for the country. It wastes some electricity but is globally cheaper. The Powerwall costs slightly over 340€/kWh but is a small unit for houses. It is guaranteed for 10 years so it will last rather 20 years. https://www.tesla.com/powerwall The utility-sized Powerpack is hopefully cheaper per kWh https://www.tesla.com/powerpack https://en.wikipedia.org/wiki/Tesla_Powerwall#Powerpack_specifications "should" cost 220€/kWh. 1.5TWh and 220€/kWh cost 330G€ every 20years or 16G€/year, not 1350G€/year. This is affordable and much less than what the inhabitants pay for electricity. ==================== The other point is that batteries are only one solution. It's a mature one, already in use at substantial scale, but not necessarily the cheapest one. I have good hope that flywheels are cheaper per stored kWh than batteries. The store-restore cycle is more efficient from night to day, less efficient over a week https://www.scienceforums.net/topic/59338-flywheels-store-electricity-cheap-enough/ and Prof. Seamus Garvey's underwater bags look cheap too and has been experimented https://www.offgridenergyindependence.com/articles/3358/compressed-air-energy-storage there are few more ideas.

This is not about B Kavanough's suitability for the Supreme Court, nor whether he is innocent or not of the accusations levelled against him ( and certainly not about D Trump ). The issue I want to discuss is whether the MeToo movement has made Due Process a thing of the past. It is no longer 'innocent until proven guilty', but guilty until you can prove yourself innocent. Have we, in our rush to provide a level playing field for victims of sexual assault, taken away the rights of a group of people ( males ) to a fair trial ? The fact that a woman, with dubious recollection ( self admitted ) of events that happened over 30 yrs ago, is to be believed beyond doubt, and could potentially ruin a man's life, unless he can prove himself innocent ( and even then, there will always be suspicions ), seems more than a little skewed. Is this the kind of 'new justice' system we need/want ? And I realize the problems with getting women to report abuse, but surely there has to be a better way than demonizing all men. Men are human and a certain number of them will do vile things. But women are only human too and a certain number of them will use this 'new justice' for their own vile ends.

No, I guess I am asking if there is any logic behind its existance, or is it just a random accident? Or can you not say for sure? For instance, the big bang has no logic behind it.

"The Universeis a self-creating, self-organizing, totally connected and strictly determinedand evolving infinite process and cycle of the construction and de-constructionof matter. Fundamental to our understanding of the Universe is the phenomena ofattraction and repulsion. By establishing the true cause of attraction andrepulsion, we gain an understanding of how the Universe works at this mostfundamental level. The result in the generation of theories that challenge manyof the theories and assumptions of Physics, establishes the cause of thenuclear and gravity forces, gives rise to a new law of Physics, and presents aprofound aspect of Human existence." The above is the introduction to an essay titled "The Materialist and Infinite Universe", which can be located by putting the title into a search engine on your computer. You can enter into discussion about what is presented in the essay on this thread. Paradigm

I am willing, just I am not allowed. However, by claiming the universe is pixel/tile/cell based I have magically provided an answer to "how" the laws of the universe are everywhere, its objective fact and its self-evident, its not my opinion. Philosophy becomes science.

That wikipedia page is only created by mathematics cranks as a means of vandalizing wikipedia.

A: Cardinality of the power set is bigger than that of ℕ . The power set is uncountable. B: Uncountability exists. A is true: "the power set is uncountable", only if B is true, that is, uncountability exists. B is true: uncountability exists, if A is true: "Cardinality of the power set is bigger than that of ℕ = the power set is uncountable". But "Cardinality of the power set is bigger than that of ℕ" does not necessarily mean "the power set is uncountable"

- Assume a natural number X such that X > all Naturals - But X+1>X - Reductio ad absurdum, no such number exists - Actual Infinity does not exist

It is a paradigm shift, its just looking at the same things but from a different perspective. As John Wheeler said: “Some principle uniquely right and compelling must, when one knows it, be also so obvious that it is clear the universe is built, and must be built, in such and such a way that it could not possibly be otherwise.” Current science (or scientists really, science cant speak) can't explain where things come from. The explanation is "well God did it", except in scientic language.

To quote John Wheeler: “Some principle uniquely right and compelling must, when one knows it, be also so obvious that it is clear the universe is built, and must be built, in such and such a way that it could not possibly be otherwise.” I trust John Wheeler, I like him. I am not sure I trust people on here.

While I lived in Canada, I made an agreement with Father Boniface. He was a Benedictine Monk.I asked him if Jesus Christ is the son of God, send a hawk to my house window after you die.After about six months after he died, I stood at my living room window in May 2012 in my house on Semmering Austria, and a hawk or falcon flew to my window and sat on my windowsill for a few seconds and then flew away. I have never seen a falcon here before and considered this a sign that Jesus Christ is the Son of God. He came to this world to save and illuminate humanity with the Christ Consciousness (I.e Sermon on the mount and the Golden rule)Praised be Jesus Christ!!Peter De-Schuster Semmering Austria

Ok, I will play God and show you how to create a universe from nothing. First step, empty space. The is no such thing as mathematics, there is nothing to count or measure. Second step, say "empty space now be composed of pixels" and your will is done. Suddenly, with one simple act, you create mathematics, we now have things to count and measure. Third step, create time. From nothing to maths in one simple step.