Re: Implementable Set Theory and Consistency of ZFC

On Mon, 29 Oct 2007 15:35:33 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

Jesse F. Hughes wrote:

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

Jesse F. Hughes wrote:

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

Jesse F. Hughes wrote:

Even assuming that there are four models rather than one, so what?
The point is that you have *not* given a proof of (5)-(8) using only
axioms (1)-(4). Thus, you have not done what you said.
Even if you show that there are seven models for (1)-(4) in which
(5)-(8) are also true, you haven't done what you said. Even if you
show there are *infinitely* many models satisfying this condition, you
haven't done what you said. So, perhaps you should either do what
you said or change your claim.

No. Because _nothing_ sensible ever counts as a proof in your conception
of mathematics.

Quite wrong. I've told you what counts as a proof of the claim that
(5)-(8) are theorems of (1)-(4). Namely, a proof of each of (5)-(8)
using only (1)-(4) as axioms.
What is so controversial about that?

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.

giggle. No argument there.

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.

Wow. I mean really, wow. You claim that 5-8 follow from 1-4, but
somehow if we add 9 then we also need to include 5-8 as axioms?

This is _incredibly_ stupid. You really have no conception of
what it means to say A follows from B.

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.

That's evidently not the only circumstance where you can't do
any sensible reasoning. I mean wow.

Han de Bruijn


************************

David C. Ullrich
.

Relevant Pages

  • Re: infinitely many nns = infinite nns?
    ... First, you would have to STATE SOME AXIOMS, ... Here, "infinitely many" is aleph-0, which is "actual infinity." ... Similarly, the "fact" that all the numbers are "finite" means that all numbers have values LESS THAN aleph-0, that the natural numbers are limited to "potentially infinite" values. ... To be honest, it won't SURPRISE me one iota if you claim that there are some natural numbers with infinitely many digits, but that they, too, have only finite values. ...
    (sci.logic)
  • Re: infinitely many nns = infinite nns?
    ... You asked for my axioms, and I really want to use the standard ones, so the question is mostly pointless, but I thought I would give clearer definitions of how I am using actual and potential infinity. ... "Aleph-0" DOES NOT EXIST outside some LOGICAL context ... "Actual infinity" is a term from informal natural-language ...
    (sci.logic)
  • Re: infinitely many nns = infinite nns?
    ... These are axioms and assumed correct. ... state whether they should be used with potential, or actual, infinity. ... cannot use potential infinity for some conclusions, ... through Axiom, mathematical induction, or as it is sometimes called, ...
    (sci.logic)
  • Re: Existence of reals and observation of them
    ... infinity, then infinity doesn't exist even as cardinality. ... You need to read the axioms and the derivations of the theorems ... if persons of good will could continue to disagree over basic ... science of mathematics. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... These infinity arguments always seem to boil down to ... standard models vs non-standard models. ... Peano's axioms are model blind: ...
    (sci.math)