Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
stephen_at_nomail.com
Date: 11/21/04
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Date: 21 Nov 2004 18:47:19 GMT
In sci.math Eray Ozkural exa <examachine@gmail.com> wrote:
: stephen@nomail.com wrote in message news:<cnp65n$ibp$1@msunews.cl.msu.edu>...
:> In sci.math Eray Ozkural exa <examachine@gmail.com> wrote:
:> : stephen@nomail.com wrote in message news:<cnmolo$1gp8$1@msunews.cl.msu.edu>...
:>
:> : It is obvious what cardinality means, and it is not a definite thing,
:> : because THIS PARADOX EXISTS. If you do not understand WHY this is a
:> : paradox this is your problem.
:>
:> : Do you accept that there is a paradox or not?
:>
:> : Yes or no?
:>
:> What paradox are you talking about? There is no paradox
:> involved with defining cardinality in terms of bijections
:> of which I am aware. I would not call it paradoxical
:> that different definitions of cardinality could be considered
:> that lead to different conclusions. All results in mathematics
:> are based upon your assumptions. Change the assumptions
:> and the results change.
: That would be mathematical solipsism. You can't define things
: completely freely in mathematics. Once you define "number" you have
: very little space to move...
: I think "cardinality" must be synonymous with "size of a set" or
: otherwise, it would be meaningless. To agree with your terminology,
: consider that I've replaced all occurences of "cardinality" with
: "size" in this exchange. Could you please answer my previous question
: in that fashion?
You have to define "size of set". When talking about infinity
size is not all that well defined. For a finite set "size"
can be determined by counting, which is the same thing as
putting the elements in a bijection with a subset of the natural
numbers. "Cardinality" has been defined. What you think
about the matter is not all that relevant, especially if you
think "cardinality" should mean something it does not.
To answer your question, if I replace "cardinality" with "size"
and use "cardinality" as the definition of "size" then clearly
my opinion does not change at all. If you want me me to replace
"cardinality" with some undefined term such as "awer" then
I cannot possibly answer the question because I do not know
what "awer" means.
Using undefined terms seems to be a trend in your posts.
It is quite simply bad mathematics. You must define your
terms in a mathematical argument.
: I'm saying that there are two approaches for measuring the size of
: sets, and in general the bijection account does not seem to be
: satisfactory to reason about the size of infinite sets. Do you accept
: that there is a paradox of the infinitely big? In other words, do you
: agree with the received view in philosophy of mathematics, or not?
What other way is there of measuring the size of a set? Are
you claiming that the fact that |E| = |N| where E is the even
natural numbers is somehow paradoxical? Even using a naive
idea of size and infinity this seems intuitive. How many
even numbers are there? Infinity, so |E|=infinity. There
are twice as many natural numbers as even natural numbers,
so |N|=2*|N|=2*infinity. What is 2*infinity? 2*infinity=infinity.
I am not claiming that this is at a rigourous, but just a
naive and intuitive way at looking at the problem.
In general subsets seem particular useless as a measure of size
because often the domains of the sets are disjoint. I suppose
you could recast everything into sets of sets, but that seems
cumbersome. A very classic problem where the bijection definition
of cardinality has immediate consequences is the fact the cardinality
of the set of all Turing machines is less than the cardinality of the
set of all functions f : N -> { 0, 1 }. Can you make this argument
using subsets or some other definition of size? The fact that
unsolvable problems exist shows us that in a very real way
there are not as many TM's as functions.
I have yet to see any paradoxes involving the infinitely big.
I have seen things that were at first unintuitive. I suppose
it depends on how you define "paradoxical". Technically
the definition does include things that at first seem wrong
but are in fact right, but people often use the word to mean
something that is contradictory. However a paradox of the former
kind should not be taken as evidence that something is wrong, other
than with our intuition.
You never did answer my question about what paradox you
are talking about.
Stephen
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