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I finally respond.
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Date Posted: 09/12/04 6:04pm
In reply to:
Duane
's message, "Finally.... Tristam meets his doom.... (bleah)" on 08/31/04 5:56am
I haven't looked into Ben's message board in a while, but here I am.
>Wade:
>
>Ugh. I've been on-and-off thinking about this, and I
>think I've finally located a deficiency in the
>argument. The deficiency is mathematical in nature,
>and I'll get to it, but first I wanted to bring
>something up:
>
>I've read most of your and Damocles's dialogue about
>this, and I've noticed that some of it focuses on
>"absurdity."
>
>
>>Wade said:
>>Is it? You be the judge. The essence of the argument
>is >that an infinite past implies an absurdity and
>therefore >cannot be correct.
>
>
>You of all people should recognize that "absurdity"
>does not equal disproof. "Absurd" means that you
>can't understand it.
No, absurd means ridiculously unreasonable, irrational, unsound etc.
>In doing some background reading, I came across the
>original statement of the paradox - Bertrand Russell
>was just trying to illustrate interesting properties
>of infinite sets. These other people then grepped his
>paradox, and wove this tangled web of crap around it,
>claiming they'd "disproven" an infinite past just
>because they can't seem to understand elementary set
>theory.
There is nothing about elementary set theory that affects the premises of my argument.
>So enough of this. On to the math.
>
>=======================================================
>== The Math Part ======================================
>=======================================================
>
>>Wade Said:
>>Because the above is a deductive argument, it can
>fail in >only one of two ways. Either the argument is
>invalid (i.e. >the conclusion does not logically
>follow from the premises) >or at least one of the
>premises is incorrect.
>
>
>The reason why Wade's paradox is incorrect is that his
>first premise is false. It's not egregiously false,
>just subtly wrong. Let's look at it.
>
>
>> 1. There is a one-to-one correspondence between
>years >passed and days passed.
>
>False.
No, it is true. I have demonstrated this.
>Here's why.
>
>Functions are simply a mapping of the elements of one
>set to the elements of another. "One-to-one
>correspondence" is a specific class of function.
>
>First, let's be clear about what is meant by "one to
>one correspondence." 1-1 correspondence is formally
>referred to as "bijection." It means that every
>element in set A maps to a single element in set B,
>that no two elements in A map to the same element in
>B, and that every element in B is "mapped to" by an
>element of A.
This is true, and I have demonstrated that it applies here:
Day 1 with year 1
Day 2 with year 2
Day 3 with year 3
etc.
Besides, you're not at all referring to my current revised version of the argument. My first premise was:
- If an infinite past is metaphysically possible, then Shandy writing his autobiography for as long as time has existed should not lead to absurdities.
>The "paradox" comes from this fact:
No it doesn't. You are using a non-current version of my argument.
>2 different infinite sets are numerically equivalent -
>they contain the same number of elements (infinity!).
>
>BUT!!! There may exist elements in one set that do
>not correspond to elements in the second set.
Not necessarily the case here, as I've shown.
Day 1 to Year 1
Day 2 to Year 2
etc.
>Huh? How is that possible? They have the same number
>of elements, but one has "more" than the other? The
>short (and correct) answer is: YES
>
>This is an axiomatic property of infinite sets. It
>arises from an interesting property of infinity - that
>infinity = infinity.
That's actually not quite true. Infinity - infinity = [answer] has, literally, an infinite number of possible answers.
>So Wade's first premise is wrong.
"Wade's" premise you are referring to does not exist in the current version.
>Here's the correct
>statement:
>
>1) There exists a one-to-one function that maps the
>elements of Y to the elements of D
>
>2) There exist elements of D that are not mapped to by
>any element of Y
Not necessarily the case, as I've shown. Challenge: give me one that doesn't have a correspondence.
>3) Therefore, there exist "days" that Tristam has not
>written about, and so he will never finish.
>
>So let's, in light of the above, look at more of what
>Wade says.
>
>>Wade Said:
>>One could deny the existence of the one-to-one
>>correspondence between years passed and days passed
>(the >first premise), but that too is ridiculous since
>we can >easily prove otherwise given an infinite
>number of years >and an infinite number of days (for
>each day that has >passed there exists a different
>year that has passed, and >vice versa).
>
>Clearly, we've seen that this is not the case. There
>exist, even assuming an infinite number of elements in
>D, elements of D that do not correspond to an element
>of Y.
Give me one specific element of D and I will be happy to show you a corresponding element.
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