The Instigator
Pro (for)
Losing
47 Points
The Contender
Con (against)
Winning
56 Points

Assuming an endless universe, there are as many Thursdays as there are days.

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 Voting Style: Open Point System: 7 Point Started: 9/21/2008 Category: Science Updated: 9 years ago Status: Voting Period Viewed: 2,645 times Debate No: 5485
Debate Rounds (3)

16 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Tatarize 6 years ago
Goya, most all of your analysis seems quite lucid. The problem is that it doesn't negate the proposition.

I fully accepted that two sets of aleph-0 items have the same cardinality namely aleph-0, even if combined. While it may be mathematically a moot point to speak of coefficients and one cold likewise argue for any (rational) coefficient (rational in that it can be stated as a fraction unlike pi or e), because you could argue for that particular bijection.

You could argue that in an endless universe, there are twice as many Thursdays as there are days, in general. The issue is that because they are all true, they are all pointless and the rather worthless. Every set of aleph-0 values has aleph-0 values regardless the nature of the set or whether one set can be said to be a subset of the other. I can easily argue for the truth of the proposition and be correct, even if it doesn't matter much because the stated proposition doesn't mean much of anything mathematically.

I freely concede that a logically sound argument that there are 7 times as many days as Thursdays, could be made along the same lines as my argument, but that doesn't mean my argument is incorrect, it means that the importance of my claim is non-existent.

In your first post there, you stated:
... = 1/3d = 1/2d = t = d = 2d = 3d = 4d = 5d = ...

Note the middle section "... t = d ....", that's the proposition and you're noting that it's true. Therefore, Pro is correct.

I freely concede the topic of the debate is rather irrelevant with regard to what we mean by infinity, and that cardinality shows that such points are rather without relevancy to anything. But, I didn't argue the topic was very important, I argued that it was true And it is true, even if that doesn't mean much.
Posted by goya1 6 years ago
again followed on from previous comment.

The reason being that by the logic of this statement, there are also as many Thursdays as months, years, decades, centuries, millennia, hours, seconds, etc. This result gives us no useful notion of the size of an infinite set. Consequently, it can only make sense to use cardinalities to determine whether an infinite set is countable or not. In other words, we can only draw the conclusion that the set of Thursdays and and the set of days each have the same cardinality, and they are countably finite. If you were to read deeper into Set Theory and Cantor's work this is what you will find.
Posted by goya1 6 years ago
This follows from my previous comment which was cut short by the word limit.

The notion of cardinality changes as we move to infinite sets. There is no sense to the statement that two infinite sets have the same number of elements. Whereas with two finite sets of the same size we can form a bijection between them to determine that they have the same number of elements, we cannot do the same for infinite sets (as shown in my previous comment).

So when we look at infinite sets we use the notion of cardinality in a different sense. Whereas previously it was used to measure the number of elements in a set, it is now used to determine what type of infinity that a set is. This may seem a strange notion, but there are in fact several different sizes of infinity.
The two simplest types are, countable infinity and uncountable infinity. A countably infinite set is one in which the elements can be listed in a line, and an example is the set of days.
Set of days = {day 1, day2, day 3, ...}
An uncountably infinite set is an infinite set which can not be listed in this way, and an example is the set of real numbers. (see Cantor's diagonalization argument for a proof that the real numbers are uncountable.)

So we have that countable infinity gives us one type of infinity, and is itself a "size" or cardinality. We also have that the test for showing two infinite sets have the same cardinality is to show that there exists a bijection between the two sets. Consequently, if such a bijection exist then there the two sets are of the same cardinality.

It should be clear that the set of Thursdays and the set of days both have the same cardinality, they are both countably infinite sets. I hope it is also clear that this is the only way that we can sensibly compare the size of these tow sets.

In conclusion, my answer, and the mathematically correct one, is that the statement "Assuming an endless universe, there are as many Thursdays as there are days" is incorrect. The re
Posted by goya1 6 years ago
There seems to be confusion on each side here regarding what exactly we mean by two infinite sets having the same cardinalities.

The message from Pro is that if two infinite sets have the same cardinality then they contain the same number of elements. In formulating your hypothesis that the number of Thursdays is equal to the number of days in an infinite time-line, you are suggesting that should we divide the number of Thursdays by the number of days we get 1. I am sure you can see that this is a silly proposition as no sense can be made of dividing two infinities. I hope that you also recognise that in dividing these numbers and getting 1 is the same as Con dividing the two sets and getting 1/7.

The above is a consequence of the fact that there is ambiguity in choosing a bijection between the set of Thursdays and the set of days. Pro, you have chosen to associate Thursday of week x to day x, yet there is no reason for doing this as one could also associate Thursday of week x to day day 2x. This would give the result that there are twice as many Thursdays as days. In fact, you could choose to have as many Thursdays to days as you wish.

As a result you have the following as true:
If number of Thursdays = t,
and number of days = d,
Then,
... = 1/3d = 1/2d = t = d = 2d = 3d = 4d = 5d = ...

One should be able to see that this result is absurd, and mathematically useless. The consequence is that there is no meaning to the notion that there are as many Thursdays as days. But Pro, please take comfort in that the same could be said of the number of Thursdays being 1/7 of the number of days.

I will now explain a briefly what Cantor meant by two infinite sets having the same cardinality, and what exactly cardinality is.

Cardinality gives the notion of the size of a set. When dealing with finite sets, cardinality is the same as the number of elements in each set, if they are the same then they have the same cardinality.
Posted by beem0r 9 years ago
Blasphemy!
Posted by Tatarize 9 years ago
They should be swayed damn it. I'm right and gave the logical proof of the statement. Not shenanigans.
Posted by beem0r 9 years ago
Just to correct one thing you said, while I did endorse that the cardinality was strictly equal, I argued that this does not allow us to say they have an equal number of members when applied to infinite sets. So I did not endorse PRO's proposition in doing that.
However, most of the rest of your comment is quite accurate.
Posted by beem0r 9 years ago
Roar.
Posted by Lightkeeper 9 years ago
Great debate gentlemen! Both propositions were well put and well argued.

I would go Pro for the reasons below.

Con's problem is that he attempts to look at infinity in a finite fashion.
"after w days (where w is infinity) have passed".... that is flawed. Since w is infinity there will never be an "after w days".
Con has conceded in the last paragraph of the debate that you can make a 1:1 cardinality for these two sets. Therefore, in his very conclusion to the argument he has in fact endorsed Pro's proposition.

One way to deal with this is the following:
1. Take out all the thursdays from now to infinity and put them in a separate set called T {1,2,3,4,5....}
2. Keep the remaining set of days from now to infinity as it is and name it D{1,2,3,4,5.....}
3. Place the two sets side by side.
4. You will see that for each T there is one and only one D.

Yes, it's hard to put your head around it. That's because us humans aren't really used to infinities ;)

I can't vote yet but if I could I'd go Pro.
Posted by Lightkeeper 9 years ago
This is an interesting debate. I can just see where both participants are going wrong. And where they are going right. This will be a tough call.
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