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:
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.

Uh, no it doesn't.


Uh, yeah, it does.

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?

I don't think so. So what?


So, define pi<4 in terms of 'e', or january<february in terms of 'e'.

<snip>
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,

That's the continuum *hypothesis*? Or did I miss something...


Apparently, you did. CH states that aleph_1=c. That is, it states that there is no aleph between aleph_0 and c.

or that omega-1=omega.

Strawman. Nobody has said that but you.


What? Review your ordinal arithmetic.

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".

Where does set theory claim "these sets are equinumerous"? I'm pretty
sure it just claims that they're bijectible.


"Equinumerous" is a term commonly used by set theorists.

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.

So? You arguments are against calling cardinality "size". Well, get
this, set theory *doesn't* call cardinality "size". It calls it
cardinality. It uses it as shorthand for bijectability. Whether you
call it size or not is totally irrelevant.


If I make statements about sizes of infinite sets and am told I am wrong because I am contradicting set theory, then that's an invalid objection.

The question here is whether one can
attribute a size to an infinite set, and if so, what is the most
appropriate method?

Appropriate for what?


Appropriate for drawing satisfactory conclusions about the relations between infinite sets.

Cardinality is very simple, relying only on set
membership and corresponding to set size, in the finite case.

Cardinality is, indeed, very simple. Amazing then how much some people
struggle with it.


That's because it's too simple to satisfy some basic intuitions.

Infinite
sets require some sort of inductive definition, some kind of order or
process.

Tosh. What inductive definition, order or process defines the set of
all functions from the natural numbers to the natural numbers?


Are you kidding? Each such function can be defined by the natural mapped to each natural, and so consists of a countably set of naturals. Where a set has size s, the number of functions from the set to itself is s!. The naturals themselves require an order relation, so defining anything based on the naturals involves order a priori.

They can't be finitely defined on an individual set membership
basis.

Is this supposed to mean "you can't name every element of an infinite
set explicitly"? Or what?


Yes, you could put it that way.

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.

Uhh. I think all infinite sets can have an (infinite number of?)
ordering(s) defined upon them, but it's totally wrong to say that all
infinite sets have "a notion of order implemented in the definition".

No it's not. Think about it.


By all means, say "they have the same cardinality", but don't use
"equinumerous" or "set size" when discussing transfinite cardinalities,

Why should everyone else change terminology for your benefit?


To avoid the objections of many to your claims that cardinality is set size for infinite sets.

because they are only very rough classifications.

Compared to what?


Compared to other approaches that yield a full spectrum of infinite sets.
.

Relevant Pages

  • Re: Anti-diagonalist page
    ... >There is also Richard Arthur (Chair of the Department of Philosophy, ... have a cardinality, and people do maybe use the phrase "cardinal ... if any two such infinite sets ... the number of naturals, and when he makes _that_ objection ...
    (sci.logic)
  • Re: Galileos Paradox and the Project of the Reals
    ... call cardinality "size" becase it makes intuitive sense to most people ... It uses it as shorthand for bijectability. ... If I make statements about sizes of infinite sets and am told I am wrong ... and so consists of a countably set of naturals. ...
    (sci.math)
  • Re: Anti-diagonalist page
    ... that axiomatic set theory just ignores I wouldn't find ... the number of naturals, except when they're speaking ... very informally - instead they say that the cardinality ... For Dedekind defined infinite sets as those that could be ...
    (sci.logic)
  • Re: Galileos Paradox and the Project of the Reals
    ... call cardinality "size" becase it makes intuitive sense to most people ... It uses it as shorthand for bijectability. ... If I make statements about sizes of infinite sets and am told I am wrong ... and so consists of a countably set of naturals. ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... inserts Peano's successor relation as well. ... call cardinality "size" becase it makes intuitive sense to most people ... If I make statements about sizes of infinite sets and am told I am wrong ... and so consists of a countably set of naturals. ...
    (sci.math)