Yes.I will try this. As n goes to infinity (n E N) n/n, does this equal 1?
If it's allowed, then the input is infinite and the output is finite. Does this solve the problem you had with my "contradiction"?
As far as I can tell, that would just define every set to have size 1. That's well-defined, but useless.
And it's not my problem with your contradiction. It's everyone's. Do you understand the necessity of having well-defined procedure for determining the size of sets? Do you understand that your procedure does not give a defined, unambiguous, and relevant result for each set?
Does what equal 1? You havent defined anything.
If it's allowed, then the input is infinite and the output is finite. Does this solve the problem you had with my "contradiction"?
You have actually no clue at all what you are doing.
How do you define the division of a set by another set? Why wiuld the result be a single number?
Clear these matters out first.
Yes.
If it's allowed, then the input is infinite and the output is finite. Does this solve the problem you had with my "contradiction"?
As far as I can tell, that would just define every set to have size 1. That's well-defined, but useless.
And it's not my problem with your contradiction. It's everyone's. Do you understand the necessity of having well-defined procedure for determining the size of sets? Do you understand that your procedure does not give a defined, unambiguous, and relevant result for each set?
If I add 1 to 0 for every element in the set of all natural numbers, I will have a number defined as the number of elements in the set of all natural numbers. In terms of a function, I think this is [as n goes to infinity (Sum of 1 from i = 0 to n)], where n E N.
So is this infinite number well-defined?
All you're showing here is that your size measuring procedure is ill-defined on the set of all natural numbers.
I feel like one could say the same thing about Cantor's diagonal argument in that the size of the reals is ill-defined when it comes to using the naturals to measure it. There are not enough natural numbers to correlate to each element of aleph 0, just like there aren't enough natural numbers to correlate to each real number.
If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.
But how can any output not be a natural number in the set of natural numbers?
I feel like one could say the same thing about Cantor's diagonal argument in that the size of the reals is ill-defined when it comes to using the naturals to measure it. There are not enough natural numbers to correlate to each element of aleph 0, just like there aren't enough natural numbers to correlate to each real number.
If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.
But how can any output not be a natural number in the set of natural numbers?
Ryan, I've tried to point this out to you before. Essentially, you've noticed a pattern that works consistently for finite sets, and then assume that the pattern must hold for infinite sets. But you haven't show that the procedure that generates your pattern must hold.
In fact, all you've done in this thread and the Cantor thread is prove, over and over, that your procedure does not work. Instead of abandoning the procedure, which you clearly have no reason to believe holds other than your intuition, you decide to abandon set theory instead! That is fundamentally the flaw in your logic. When your reasoning, given an assumption, leads to a contradiction, then you reject the assumption!
If aleph 0 is truly all of the natural numbers, and is truly a number, then let's abbreviate it as the number AN.
Matching the nth natural number element to every consecutive n: 1 to 1, 2 to 2, 3 to 3 ... AN to AN.
What is so bad about calling AN a natural number, and specifically the last natural number? Why can't we start on the other end of the natural numbers? If we try to add 1 to it, then we will just end up with AN + 1 which isn't a natural number.
We know that the natural numbers end because we know that there are larger infinities. Why is this such a problem?
I feel like one could say the same thing about Cantor's diagonal argument in that the size of the reals is ill-defined when it comes to using the naturals to measure it. There are not enough natural numbers to correlate to each element of aleph 0, just like there aren't enough natural numbers to correlate to each real number.
If you require that the only legal outputs are natural numbers, you can't complain about problems if you input a set that doesn't have size equal to a natural number.
But how can any output not be a natural number in the set of natural numbers?
Ryan, I've tried to point this out to you before. Essentially, you've noticed a pattern that works consistently for finite sets, and then assume that the pattern must hold for infinite sets. But you haven't show that the procedure that generates your pattern must hold.
I admit, I don't have a complete understanding of set theory. But there seems to be similar properties that infinite sets and finite sets share even in set theory:
- A finite number of elements and an infinite number of elements can be contained in a set.
- There is a number that represents how many elements are in a finite set, and there is a number that represents how many elements are in an infinite set.
- Both finite and infinite sets are countable using natural numbers (at least the infinite set that is being discussed).
- Both finite and countably infinite sets have sets larger than them.
There are many similarities.
Please see Post #24, and please comment on it if you can.
In fact, all you've done in this thread and the Cantor thread is prove, over and over, that your procedure does not work. Instead of abandoning the procedure, which you clearly have no reason to believe holds other than your intuition, you decide to abandon set theory instead! That is fundamentally the flaw in your logic. When your reasoning, given an assumption, leads to a contradiction, then you reject the assumption!
I am trying to do what is allowed in set theory to show a contradiction. I may not be doing a very good job, but I am working as hard as I can with what I have.
