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

so, obviously you have not learnt anything!

20 hours ago, Genady said:

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

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

22 minutes ago, ahmet said:

so, obviously you have not learnt anything!

Moderator Note

Posts should be about the topic, not personal attacks.

You’ve been warned about this before, so you obviously haven’t learned anything about that.

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.

Edited 4 hours ago4 hr by Genady

28 minutes ago, Genady said:

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.

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

Edited 4 hours ago4 hr by studiot

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.

28 minutes 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.

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

Both disciplines suffer from a plethora of notation, this may help.

1 hour ago, ahmet said:

whatever someone says, you will eventually suppose that you were right in every case. May I suppose that this was the "approach/supposition of stupid English Society"?

anyway, what I said was exactly: that OP was wrong in his/her assertion. studiot is either.

but as said, whatever we say, I am sure that someone will suppose that they were always correct. So, what to discuss here?

nothing. And please accept that this is bigotry.

If you wish to contradict someone, my expectation would be to follow the bald negation with a proper mathematical substantiation of your own claim.

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/ 🙂

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.