Re: The modern mathematical concept of infinity is indefensible

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

Jesse F. Hughes wrote:

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

See? There are axioms and there are theorems. The axiom is that infinity
cannot exist, as a completed entity. The theorem is that there is no Big
Bang. Just an axiom and logic.

Right. Your opinions about infinity are axiomatic. That is, they are
unjustified assumptions.

Empirically justified assumptions, that is.

You have no empirical justification for your rejection of
singularities. You reject them because they involve infinity, but you
claim infinity is impossible because it doesn't occur in nature.

I've never seen a critter living on another planet and thus, I
conclude, all living beings are on earth. Now, I know that there is
some (inconclusive) evidence of rudimentary life on Mars, but there
can't be life on Mars, because I've already concluded all life is on
earth.

You don't see the circularity?

I think we all agree on that point.

In case you didn't know: axioms come to us from outside. The good axioms
aren't made up from thin air. Where do you think the axioms of Euclidian
geometry originally come from ?

That's fine by me. But from now on, let's describe your rejection of
infinity accurately: it is an unjustified assumption.

--
"Clouds are always white and the sky is always blue,
And houses it doesn't matter what color they are,
And ours is made of brick."
-- A new song by Quincy P. Hughes (age 4)
.

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. ...
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  • 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)