Re: Another Inconvenient Truth

On Aug 8, 8:30 am, step...@xxxxxxxxxx wrote:
Why do you call it "wrong"? To what are you comparing it to determine
that is is wrong? What objective standard do you have access to that
disagrees with set theory?

Stephen

One topic that comes up frequently in set theory threads here in this
newsgroup is physics. In particular, HdB, like many nonstandard
mathematicians, argues in favor of replacing the Axiom of Infinity in
ZF
with its negation because one cannot prove that infinite sets exist in
the physical world.

The standard mathematicians reply by stating that it is still possible
that there exist infinitely many particles, and even if they don't
exist
infinite sets can still exist in the Platonic realm of mathematics
independent of any physical reality.

We notice that no one, not even a nonstandard mathematician, has ever
initated a thread declaring that "2+2=5" or any other formula whose
negation is an theorem of ZF-Infinity, because such sentences are
verifiable in the physical world. It is only those formulas which are
provable in ZFC but not in ZF-Infinity which are challenged by HdB and
other nonstandard mathematicians.

The usual response of standard mathematicians is that the usefulness
of
the Axiom of Infinity is not in the existence of infinitely many
particles, but in the existence of the complete ordered field. Many
problems in physics can be modeled by differential equations, such as
the heat and wave equations. The nonstandard mathematicians may point
out that spacetime may be quantized (Planck length/time, etc.), but
even so, differential equations are in general easier to solve than
difference equations. This is the main reason why most standard
mathematicians defend the Axiom of Infinity.

.

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