Genady

Check this exercise:

I think, the text is mistaken. In the category of sets, a direct product of A and B is their Cartesian product C, and the morphisms are maps from C to A and from C to B, which are not injective, i.e., monomorphisms, but rather surjective, i.e., epimorphisms.

OTOH, a direct sum of sets A and B is their disjoint union C, and the morphisms are maps from A to C and from B to C, which are not surjective (epimorphisms), but rather injective (monomorphisms).

Do you agree?

Today I've found a simple example of a morphism which is monomorphism, i.e., one-to-one, and epimorphism, i.e., onto, but is not an isomorphism, i.e., does not have an inverse.

58 minutes ago, Genady said:

Can you come up with a similarly external definition of "onto" mapping?

Here it is:

Mapping [math]f: A \to B[/math] is onto if for any X and any mappings [math]p: B \to X[/math] and [math]q: B \to X[/math], [math]p \circ f = q \circ f \Rightarrow p=q[/math].

TIL that "one-to-one" mapping between two sets can be defined as an external property of the mapping, i.e., without any reference to elements of the sets and to what happens to them under the mapping. Here we go:

The mapping [math]f: A \to B[/math] is one-to-one if for any X and any mappings [math]p: X \to A[/math] and [math]q: X \to A[/math], [math]f \circ p = f \circ q \Rightarrow p=q[/math].

Can you come up with a similarly external definition of "onto" mapping?

16 minutes ago, swansont said:

You need to provide a citation

It's from fb: https://www.facebook.com/permalink.php?story_fbid=pfbid028dPTEhRvxf6j3HMNy52Ymv742GMuQXekebvDo8DzjkAU9EuiENkDdM1Kt2TAUM8hl&id=100090706310719&__cft__[0]=AZZHIbs76gT7OBtsHj6mMXIx5moXtwff6b04aSxChSwB5lEXYpPMSGOGKRrOS9uci6M4uNynPdHRcXUqgAn7915tu-8VZC_TjOrw2r4iP4O92OMOlQ6FBa5PGInZyi69m869_BZmfljXXH2ce7DMOVWjbE4olX7ZzZg8C7glt3qkEg&__tn__=%2CO%2CP-R

10 minutes ago, DimaMazin said:

anglegenta and sectorsgenta

These are not words in English.

32 minutes ago, Otto Kretschmer said:

Does it? The quality of politics around the world suggests otherwise...

Yes, it does. It is an effective manipulation technique.

Because it works.

I remember getting a question on my US citizenship exam (many-many years ago), "What is Constitution?" with one line for an answer. I've answered, correctly, "Constitution is the supreme law of the land."

On 3/26/2026 at 9:59 AM, Genady said:

  On 3/26/2026 at 8:52 AM, studiot said:

a plethora of notation

This notation,

(rather than, e.g., S or s) is new to me.

Only two days ago it was new to me, and it is already in my next book:

I've read it only last month and they've made a movie already 🙂

A good math joke, albeit 30 years old:

1 hour ago, studiot said:

Yes I agree they are the same construction with different notation.

This is exactky what I mean by a plethora of notation.

Right. And my book says,

but still, since all the numbers here are integer, the definitions

[math]|\lambda(m+n)-(\lambda(m)+\lambda(n))| < M_{\lambda}[/math]

and

[math]\left\{ \lambda(m+n)-(\lambda(m)+\lambda(n)) \right\} \,\text{is finite}[/math]

are equivalent.

4 hours ago, Genady said:

The construction that I've learned recently follows closely the "2.12. Schanuel (et al.)’s construction using approximate endomorphisms of Z ([2, 11, 16, 29, 30, 1985])" in your first linked paper.

Interestingly, my book cites rather "Norbert A’Campo, A natural construction for the real numbers, Elemente der Mathematik, vol. 76 (2021)."

P.S. Ah, I see that A'Campo's is your second linked paper. Perhaps, there is some difference that I didn't see yet.

I still don't see a difference between the two constructions mentioned above. The first says,

The second,

Does anybody see how they are different?

35 minutes ago, studiot said:

You don't actually have to post in a thread to 'vote', which is why I originally responded as I did for I did not then know who it was.

The mods can see who voted what, of course.

Whoever it was that downvoted you, I've balanced it by an upvote, just as you've done for me earlier.

1 hour ago, studiot said:

Judging by this and your other recent threads you are following some scheme connecting formal logic and maths.

Yes, I'm studying this book:

1 hour ago, studiot said:

a plethora of notation

This notation,

(rather than, e.g., S or s) is new to me.

1 hour ago, studiot said:

One note of interest is confirming that the 1 in the naturals is the same as the 1 in the reals or the integers

This note of yours has a direct relevance to this "Quiz" of mine here: https://www.scienceforums.net/topic/140398-from-naturals-to-integers-quiz/ 🙂

47 minutes ago, studiot said:

There has been continued interest in this subject since the days of Cantor and Poincare.

Here are a couple of recent papers.

One note of interest is confirming that the 1 in the naturals is the same as the 1 in the reals or the integers for those who widh to be strictly pedantic.

Realnum_RMJ-2015-45-3-737.pdf acampo-real.pdf

The construction that I've learned recently follows closely the "2.12. Schanuel (et al.)’s construction using approximate endomorphisms of Z ([2, 11, 16, 29, 30, 1985])" in your first linked paper.

Interestingly, my book cites rather "Norbert A’Campo, A natural construction for the real numbers, Elemente der Mathematik, vol. 76 (2021)."

P.S. Ah, I see that A'Campo's is your second linked paper. Perhaps, there is some difference that I didn't see yet.

2 minutes ago, studiot said:

Yes all constructions of R are isomorphic, I think at the current count there are more than 10 different ones.

The only real issue about this is do you include the number zero in the naturals ?

I don't see the downvote as justified, so I have added a balancing +1

Thank you!

I didn't know about 10 different ones, only about three, I think. And they all constructed rational numbers before constructing reals. So, a direct route from Z to R without Q was interesting.

Real numbers can be constructed directly from integers, without a construction of rationals.

Given natural numbers [math]\mathcal{N}=\left\{0, 1, 2, ... \right\}[/math],

Why do they identify [math]x \in \mathcal{N}[/math] with [math]\left\langle 0,y \right\rangle[/math] rather than [math]\left\langle x, 0 \right\rangle[/math]?

Does it matter?

If yes, how / when?

2 minutes ago, KJW said:

Just to clarify, the definition says, "for each y in S", which includes x₀, whereas y must not be equal to x₀.

Right. As they say in the first part of the definition, xRy and x=y are mutually exclusive.

3 minutes ago, KJW said:

I think I see it: y has to be not equal to x₀.

That's it!!

4 minutes ago, KJW said:

@Genady, is the mistake you see that subset S requires at least two elements and not merely be non-empty?

Ha, that's too. Still, there is another one, related.

Without going into any technical details, the short answer is,