Re: Implementable Set Theory and Consistency of ZFC

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Jesse F. Hughes wrote:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Nothing. I've done just _that_ in my article.
Weird. How come no one else can recognize that you've done that?

You forget that 'sci.math' is not "no one else". It's a relatvely small
Internet Cafe, with some really weird inhabitants.

Is there someone else you *have* convinced? Well, no matter. Let me
instead say "no one else of whom I'm aware". And if there *is*
someone who agrees with your claim, by all means let me know.


Also, if you *had* done that, then axioms (1)-(4) + (9) would be
sufficient for ZFC. Don't you find it a touch odd that no one else
has noticed this fascinating fact?

The fact that Infinity X is an axiom of standard ZFC makes it necessary,
it seems, to include the axioms (5-9), in order to make infinite sets
make more "look alike" finite sets. Which wouldn't have been necessary
with a more realistic approach (I mean, e.g. Choice is provable within
the realm of finite sets, as is well known).

You *do* know that if (1)-(4) prove (5)-(8), then so do (1)-(4) + (9),
right?

I can't do any sensible reasoning with Infinity (9 = X right?)
included.

It is an well-formed formula, yes? And thus, we can consider a theory
in which (9) (or X, whatever) is an axiom. So if you can't do any
sensible reasoning with (9), it appears to be a personal issue.

--
"[Sometimes, I don't know what to do] so I guess. Technically what I
do is called making a hypothesis. It's like what physicists do, and I
basically operate a lot like a theoretical physicist."
-- James S. Harris
.

Relevant Pages

  • Re: Reals without infinity
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  • Re: Ultimate debunking of Cantors Theory
    ... > THE ONLY SETS WHICH EXIST ARE FINITE SETS. ... The advantage for the poster is that Cantor's diagonal construction of the Reals doesn't exist, or indeed any form of the diagonal argument applied to infinite sets. ... Indeed, if you consider that the existence of more than one type of infinity to be "absurd", you can use this to show that the Axiom of infinity must be false through a reductio-ad-absurdum argument. ...
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  • Re: Why does everyone do it?
    ... The set w + w satisfies the axiom. ... merely theorems, not axioms, if ZFC were restricted to finite sets? ... Infinity + ~Infinity. ...
    (sci.math)
  • Re: Why does everyone do it?
    ... Any set theory with unrestricted comprehension is inconsistent, ... if it does not have the axiom of infinity. ... of a set theory with finite sets only. ...
    (sci.math)
  • Re: Some basic set theory questions
    ... with me that getting the existence of the empty set from ... using infinity. ... This debate concerns whether the Axiom of Infinity is ... required to prove that the empty set exists. ...
    (sci.math)