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Re: a formal construction of Dedekind cuts
Posted:
Feb 27, 2013 4:14 PM


In <Zbdnc8kkPww2rPMnZ2dnUVZ_vednZ2d@giganews.com>, on 02/27/2013 at 11:51 AM, fom <fomJUNK@nyms.net> said:
>Perhaps the best apology (explanation) concerning >the nature of mathematical logic on my bookshelves >is Veblen:
The issue is not the nature of mathematical logic, but the nature of your text. You did not do what was described by the authors you quoted. In particular, you neither defined the terms you used nor provided axioms referring to them.
>One thing that has amazed me on sci.math and sci.logic >are the professional participants who do not even seem >to know what constitutes the mathematics about which >they pontificate.
Extraordinary claims require extraordinary proofs. An alternative explanation is that they understood and you didn't.
>The following discussion of the real number system
As before, what the authors wrote is not what you were doing.
>I included the final two remarks because of specific >unwarranted "corrections" I have received on these >newsgroups.
Please cite an unwarranted correction or a correction inconsistent with what you quoted.
>As for other criticisms concerning a somewhat >terse presentation. The following is from "A >Theory of Sets" by Morse.
Please compare it with your text.
>The phrase "finished class" is taken from the >history of Cantorian set theory
A formal construction has nothing to do with history.
>No. What I did in this post was not extraordinarily >awful.
I'm afraid that it was.
>It is just that no one even thinks about what it actually might >take to construct the real numbers within a theory of classes.
Except perhaps the students in Analysis 101 or Set Theory 101.
>There is no "description theory" in the language >of set theory by which the ordered relations >required to formulate "models" may be formed.
There is no group theory or topology either, which doesn't prevent defining groups, models and topologies using set theory.
>I formulated an idea of how the Dedekind cuts might be formed if >someone was actually "working in ZFC".
No; to do that you would have needed to use the language of ZFC, not invent your own. You also managed to make it look much more difficult than it actually is.
>The theory of real numbers is not the same >as the theory of transfinite numbers.
Water is wet. Why would anybody believe that they are related, much less the same?
>And, there are obvious problems with the definiteness >of infinite ordinals since forcing models can manipulate >cardinalities.
I'm not sure what you mean by "definiteness", but cardinality is defined by what maps exist. Sets that are equipotent in a given theory need not be equipotent in a model constructed using that theory.
>Thus, the "great questions" of set theory preclude the transfinite >sequence from being immediately interpretable as real numbers.
Or as bananas. Neither makes any sense.
>Nor are the real numbers urelements,
ZFC doesn't have urelements.
>The identity of real numbers is obtained by the ordering of the >natural numbers retained through each step of a formal >construction.
Is that supposed to mean something? If you don't want to define the real numbers axiomatically, there are several simple constructions. All of the standard constructions define a real to be a single set, so there is no issue of how to define identity.
 Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
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