Re: Logarithm of transfinite numbers

Tony Orlow wrote:
MoeBlee said:
Tony Orlow wrote:
Actually, I can axiomatically state these things and treat Big'un as a
primitive, and if I can derive no contradictions, and can derive useful
results, you have nothing to complain about.

That would be good. But since you criticize other mathematics on the
basis that its axioms aren't true as statements about a fundamental
reality, then your own axioms are subject to such scrutiny too. I don't
know why you would think it is so manifestly true that there is an
object that exists as a fundamental reality that is the length of the
real line but (if you do hold this:) that there isn't an object that is
the set of counting numbers.

Yes, that's not a vacuous point, and one I can appreciate. Assuming a length to
the real number line when it has no discernible ends does seem somewhat
arbitrary, and that's why it needs to be assumed a priori, since it's not
really a derivable value.

The question is why you think that is a priori true as opposed to the
axioms of set theory.

But, if we say there exists this infinite line, then
there is SOME length to it which is some infinite value, and we simply name
this value Big'un, and see what we can do with it.

I don't see that as being a priori true any more than the axioms of set
theory.

It appears that the
assumption that this value is not only the length of the line, but also the
number of points within any unit segment of it, actually fits perfectly with
the Inverse Function Rule and answers the question of the relationship between
the naturals and the reals, without producing any contradictions that I've been
able to detect.

"Fits" reflects a coherence theory of truth, as opposed to the
criterion of fundamental truth you've challenged set theory with. Also,
lack of contradiction reflects an approach that has a longer track
record with set theory than with your own notions, especially since
your own notions can't even be looked at for formal consistency until
they are formalized. And you'd have to say what is "the question" as to
"the relation" between the naturals and the reals you have in mind.

Oh. Define 0.

The unique x(Ay ~yex).

That sounds like a definition of the null set ala von Neumann,

It's basic to Z set theory (though Zermelo's original paper did take
the existence of the empty set as axiomatic since he hadn't formalized
to the extent of making the first order logic explicit). And, if I'm
not mistaken, it goes back even much further in the past than Zermelo.

but not
necessarily going to the heart of what 0 is.

What 0 is? It exists as an observable object? As a platonic object?
Whatever it is, for me, the above definition captures a splendid
representation.

Really, I was referring to its use
in the Peano axioms, where is is taken as a primitive.

Then since you know it is primitive in first order PA, I don't see the
sense of asking for a definition of it in first order PA.

I see no reason why it
can't be taken as an assumed primitive and a starting place, and actually see
this as a natural starting place for mathemtics as a whole. Start with nothing.

Fine. But it turns out that if you have an axiom that says that for any
set, there is any subset of members having a certain property, then 0
as a primitive is redundant.

MoeBlee

.

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