Re: Galileo's Paradox and the Project of the Reals

Mike Kelly wrote:
Tony Orlow wrote:
Mike Kelly wrote:
Tony Orlow wrote:
Mike Kelly wrote:
Tony Orlow wrote:
David Marcus wrote:
Tony Orlow wrote:
David Marcus wrote:
You haven't noticed that Tony is a crank? And, that what he wrote above
is vague nonsense?
I'd rather be a crank than a cog, or a stick in the mud, David.
Then I guess you've got your wish. Tell me: are you incapable of
learning math or do you choose not to?

I choose not to accept nonsense conclusions justified by unsound axioms,
if that's what you mean...
Which axioms would those be?

Take the axiom of infinity, for instance. It declares an infinite "set",
but it cannot do so with just the notion of set membership, so it
inserts Peano's successor relation as well.
Please state the axiom of infinity.
There exists a set such that 0 is an element of the set, every element
has a successor, and the successor of every element is also an element
of the set.

What is 0? What is "successor"? What does this have to do with Peano's
successor relation? How does the axiom of infinity not rely solely on
set membership?


Because it orders the sequence linearly, not as an unordered set.

There are no pure infinite
sets. All infinite "sets" are sequences or other inductive structures,
with order.
I have no idea what your definition of "a pure set" is. Pray provide a
definition.

One using only "element of" in its definition, without using succ() or '<'.

Succ() or < can be defined in terms of "element of" so apparently all
sets are pure sets.


Is pi an element of 4? Is January an element of February?

The axiom of choice is highly suspect, if not in what it actually
states, then in its application. The axiom of induction is too limited
by the faulty conclusions of limited induction.

You cannot divide a ball into five pieces and reassemble those pieces
into two solid balls of the same size as the original,
Not in real life, no. It's hard to cut non-measurable slices of a ball
by hand.

It's hard to measure non-measurable pieces too, but I guess the Axiom of
Choice somehow gives you an out.

Not sure what this is supposed to mean. Anything?

where measure is not ignored.
Private use of the word measure again, I guess.

Right, like your use of the word "non-measurable".

No, I was using the standard meaning of "non-measurable". The one that
you claim doesn't exist!


The set of even naturals is non-measurable, but it has a measure relative to the naturals, so I am speaking of a broader meaning of the word.

I don't need to accept proper subsets being the same size as their proper supersets.
Set theory doesn't say proper supersets are the same size as their
proper supersets. Set theory doesn't talk about "size" as far as I
know.
Cardinality is an extension of finite size,

Which theorem or definition of set theory refers to "finite size"?

such that it is referred to as set size.

Which theorem or definition of set theory refers to "set size"?

You STILL seem to be confused between how people describe set theory
and what it actually says.


You don't seem to have a clue as to the rationale behind it.

Set theory says there are sets that can be bijected with some of their
subsets. Do you accept that?

Sure.

So you're just arguing about terminology. You accept what set theory
actually *says*, which is that some sets can be bijected with their
subsets. And, for example, that the evens can be bijected with the odds
and both can be bijected with the naturals.

I don't argue that. I argue against thinks like c=2^aleph_0, or that omega-1=omega.


Using set theory doesn't require calling cardinality "size". People
call cardinality "size" becase it makes intuitive sense to most people
to think of it as size. I must've told you a dozen times that if you
wish you can just replace "has equal cardinality to" with "is
bijectible with" and nothing is changed. You don't have to "accept
proper subsets being the same size as their supersets" because set
theory doesn't say that.

It certainly claims to have the correct answer when it says these sets are "equinumerous".


I'd like an acknowledgement from you that set theory doesn't say
anything about "size". You've been told this lots of times but keep
repeating the same garbage.


The axioms don't say "size" but that's clearly the motivation for transfinite cardinalities. The question here is whether one can attribute a size to an infinite set, and if so, what is the most appropriate method? Cardinality is very simple, relying only on set membership and corresponding to set size, in the finite case. Infinite sets require some sort of inductive definition, some kind of order or process. They can't be finitely defined on an individual set membership basis. So, when it comes to infinite sets, this notion of order, however it is implemented in the definition, should be considered in relative "set" measures.

By all means, say "they have the same cardinality", but don't use "equinumerous" or "set size" when discussing transfinite cardinalities, because they are only very rough classifications.
.

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