Re: Logarithm of transfinite numbers

Tony Orlow wrote:
I think that, as an assumption, it is not inherently wrong, but sensible,
thatan infinitely long oline has an infinite length. The only leap is assigning
some particular infinite value to this length and calling it a unit, but units
are like that, arbitrarily declared and then built upon.

You say that your system is based on fundamental truths that set theory
is not based on. Yet, your axioms are turning out to be no less
arbitrary than those of set theory.

It doesn't seem to you that the infinite number line has an infinite length, or
that we can put a name on this infinite length?

Define 'length' and we can discuss. On the other hand, since infinity
is not something we can look at with direct observation, what infinite
things have or don't have must depend on an axiomatization and
definitions of the primitives. I don't know of any primitive notion
that demands that there be an object that is the entire length of the
real numbers under the standard linear ordering of the reals.

"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.

The question is what the relationship is, formulaically, between the discrete
naturals and the continuous reals, if any such formula is possible. Standard
theory holds that c is 2^aleph_0, which may be aleph_1, right? That is based on
a combinatorial approach that confuses infinite languages with infinite
quantitative sets.

You keep saying "confuses" and things like that. Set theory has
primitives, axioms, definitions, and theorems. There is no confusion.
You still haven't even started thinking about the point I've been
making for months now - definitions are not substantive.

If we can map the naturals to the reals in the unit
interval, then we've created a bijection between the two, akin to that proposed
by reflecting the bits of the binary naturals around the binary point to create
the binary fractions. This bijection is generally dismissed because the binary
naturals contain no infinite strings, but if we're talking about the
hypernaturals, that objection is moot, and the bijection stands, and the
unbounded megabigulous discrete set is equal to the bounded megabigulous
continuous set of reals in the unit interval.

Get back to me when you've defined all those terms.

In addition, the whole question
of the Continuum Hypothesis is answered,

If your theory is consistent and rich enough to express basic
arithmetic, then there will be other undecidable sentences, even if
not the continuum hypothesis.

since not only are there a full
spectrum of formulaic infinities between oo and 2^oo, but there are a whole
spectrum of infinities less than any standard oo as well. Are these not
questions unanswerable in standard set theory?

It is undecidable in set theory whether there are cardinalities between
omega and 2^omega. If your theory is consistent and rich enough to
express basic arithmetic, then there will be undecidable sentences in
your theory. As to infinities less than omega, no, in set theory it is
not undecidable; in set theory there are no infinities less than omega.

Okay, then von Neumann didn't pioneer the confusion of a set with the size of
the set, he just formalized it.

Von Neumann was among many set theorists provding formalizations,
definitions, and theorems. You have no comprehension whatsoever of the
history of set theory.

0 is the relative here and now, the origin, the reference point for all
quantity. Disagree?

How can I disagree with what you don't define - 'origin', 'reference
point', 'quantity'.

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 didn't. I suggested I was using it as a primitive.

You asked Virgil (or whoever it was) to define '0'.

It can be described
qualitatively, as all primitives probably should for ease of comprehension,

I have no idea what you mean by "describing a primitive
quantitatively".

but
the axioms ultimately define the behavior of the object, no?

Informally put, that's okay. Right, it's the axioms that tell you how
the primitives are going to "work" with one another.

Perhaps axiomatic statements regarding 0 can make that axiom redundant, eh?

I don't know why you say that. I said nothing like that. I said that
from the axiom schema of separation, we can prove that there exists an
object that has no members. From the axiom of extensionality, we can
prove that that there is at most one such object. Thus, taking '0' as
primitive and/or having an empty set axiom is superfluous.

I
can't really tell from this description of the axiom in question.

Cf. Chapter 1 of just about any good set theory book. And, as I recall,
I've mentioned it at least a few times in posts to you. From the axiom
schema of separation and axiom of extensionality, we can prove the
existence of a unique empty set.

MoeBlee

.

Relevant Pages

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