Shandean Postscripts to Politics, Philosophy, & Culture - Wittgenstein's 'cancerous growth': An Incident in the Philosophy of Mathematics:

From: monacojerry Date: March 3rd, 2005 02:53 pm (UTC) Re: Exchange at real_philosophy in re Witt 'cancerous growth (Link)

jeffrock
2005-03-02 23:47 (from 142.22.16.52) (link) Select
If you mean to say that there are more members in an infinite set than in another infinite set, I am going to accuse that person of tomfoolery with numbers.

Then you haven't fully understood Cantor's argument. (If you think you have, please point out a hole in his reasoning.)
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sacundim
2005-03-03 02:25 (from 209.204.158.254) (link) Select
It's not confusing me.

Ok. Amendment: "It pisses off people like Lane."
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i_am_lane
2005-03-03 03:50 (from 70.112.97.254) (link) Select
*sniff*

That's better.

:)
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bvihvhkjbkfdmek
2005-03-02 22:33 (from 213.78.21.196) (link) Select
What it buys us is convenience (as I said, "size" is a shorter word than "cardinality"). Also, insofar as the English word "size" has any definite meaning at all, it seems perfectly sensible to me to define the size of a set as its cardinality, for consider this:

One method of comparing the sizes of two real objects (e.g. the lengths of two pieces of string) is to arrange them adjacently in some kind of canonical configuration (e.g. straight lines, for bits of string) and see which one is able to obscure the other. By Zermelo's theorem, all sets can be 'stretched out into straight lines' (I mean well-ordered), and so this is actually a very close analogue of the situation in set theory.

The main disanalogy is the fact that if sets A and B are both infinite, then it may simultaneously be possible for A to obscure B 'with room to spare' and vice versa (e.g. Q obscures Z with room to spare, but given a bijection between Z and Z x Z we find that Z also obscures Q with room to spare). However, by the Cantor-Schröder-Bernstein theorem, this isn't such a terrible pathology - it only happens when two sets are equicardinal.

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Context-dependence
bvihvhkjbkfdmek
2005-03-02 23:06 (from 213.78.21.196) (link) Select
To be fair, the appropriateness of "size means cardinality" depends on the context.

At one extreme, if we think about sets purely in terms of the category of sets, then the cardinality of a set is actually its one and only property.

However, elsewhere in mathematics there are all kinds of other ways that it can come to be useful for us to abbreviate a technical concept with the word 'size'. The most obvious is in measure theory.

In practice, this never causes confusion amongst mathematicians - only overenthusiastic lay people...

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Re: Context-dependence
sacundim
2005-03-03 03:25 (from 209.204.158.254) (link) Select
I think the confusion happens when mathematicians talk about mathematics with lay people. If you get two mathematicians together, they just have too many assumptions in common to realize how odd those assumptions are, and all the alternatives that get filtered out.

I can say that I've seen plenty of times when mathematicians explain some notion to lay people, and make a claim that the laypersons find unintuitive ("the set of rational numbers is the same size as that of natural numbers"; "anything follows from a contradiction"), and, faced with the audience's skepticism, adopt a completely dogmatic attitude towards it.
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