You are talking about “infinite sets” and yet you use finite sets, {1 … 10} and {1 3 5 7 9} as examples. That is an obvious error. You are correct that the two finite sets cannot be put into a one-to-one correspondence; the same does not apply to the infinite sets of natural numbers and odd numbers.But its not logically acceptable for there to be equal infinite sets of both 1-10 and set 1 3 5 7 9, AND for there to be equal numbers in both series.
He has already agreed to it, though I had to use character strings to help, {nmd, sgqdd, ehud, rdudm, … }.It has been explained n times why there is a 1-1 correspondence between {1,2,3,…} and {1,3,5,…}.
I admit that your character-string is right and contradicts my argument.You are talking about “infinite sets” and yet you use finite sets, {1 … 10} and {1 3 5 7 9} as examples. That is an obvious error. You are correct that the two finite sets cannot be put into a one-to-one correspondence; the same does not apply to the infinite sets of natural numbers and odd numbers.
Have you worked out the rule for my character-string example yet? Doing so will help you understand.
rossum
thi…I admit that your character-string is right and contradicts my argument.
However, my argument was not about a finite number of sets as you just said, but about an infinite number of sets {1-10} and {1 3 5 7 9}, each step having a one to one correspondence, but there forever being 5 more numbers in each set of the first series. So my conclusion must be that infinity just doesn’t make sense and our minds are to limited to make sense of it
It does, in fact my character string example explicitly contradicts your argument. You have agrees that it does, and so you have lost here. My set of character strings was derived using a simple Caesar shift encryption. Just move each letter in my strings forward one in the alphabet:I admit that your character-string is right and contradicts my argument.
T…The next set would be {11-10}. Why are you now saying that these are uncountable infinities when before you said that they were countable?
Correct. Every natural number n corresponds with a single odd number 2n-1.Here is the bottom line: if there is a one to one correspondence between all the odd numbers and all the whole natural numbers, then there is a ONE to ONE correspondence, and nothing missing.
Above you said “all the odd numbers” so you have no more odd numbers to add. If you add a single non-odd number, say 4, then you no longer have the set of odd numbers, but a different set.Therefore if I add one single number to the odd numbers
That correspondence can be shifted by 1, so leaving a space at the beginning to insert the additional number, 4 in my example.that set will be greater than the whole natural numbers, because there is NO NUMBER for it to correspond with, since there is already a complete one to one correspondence.
It was an attempt to get you thinking in terms of sets, rather than in terms of numbers. This concept needs to be seen in terms of sets and their elements.I am pretty sure that the “Caesar shift encryption” is just blurring the regular 1 2 3 4 5 and to infinity set, and not one of its subsets.
What? No. You can’t add 4 to the set of all odd numbers. 4 is not an odd number. If you added 4 to the set of all numbers, you would have “the set of all odd numbers, and 4” The set of all odd numbers, and the set of all odd numbers and 4 are completely different sets.If there is a one to one correspondence between all the odd numbers and all the odd plus even numbers, every position is lined up, every single number is paired. So adding an even number to the odd set would make the odd plus (say) 4 greater than the odd plus all even numbers.
I think this is an area of math were mathematicians are willing to be irrational but some reasons
We have repeatedly shown that “all the positive integers” and “all the odd integers” can be lined up in a one-to-one correspondence.If there is a one to one correspondence between all the odd numbers and all the odd plus even numbers, every position is lined up, every single number is paired. So adding an even number to the odd set would make the odd plus (say) 4 greater than the odd plus all even numbers.
I think this is an area of math were mathematicians are willing to be irrational but some reasons
No, I have shown that there are two different one-to-one correspondences between two different pairs of sets. The first correspondence is between the positive integers and the odd integers. The second correspondence is between the positive integers and the odd integers with four. The second set in the correspondences are different, so the correspondences are also different.By shifting the numbers around as you have you have admitted there is no one to one correspondence.
An interesting question, and one I am not qualified to answer. I suspect that the difference is itself an uncountable infinity, but I am ready to be proved wrong.By how much is an uncountable infinity greater then a countable infinity?
But the new set is still a countable infinity. There is more than one set with a countable infinity of elements. For example, the set of all possible words is a countable infinity.So one number added to the odd numbers make it a greater series then it was.
Not in terms of the set algebra we are discussing here. C{odd + even} = C{odd} is the correct answer, where C{…} is the Cardinal Number of the set. As I said, this problem involves set algebra.Therefore the odd plus even numbers are greater than the odd numbers.
If you subtract a countable infinity of numbers from an uncountable infinity of numbers, you are left with an uncountable set.By how much is an uncountable infinity greater then a countable infinity?
Here is an example of two different 1-1 correspondences between the sets A and B.I don’t see how there can be two different one-to-one correspondences IF one to one correspondence means there are only pairs between the two sets…
They have to be different. The first correspondence is between set A and set B. The second correspondence is between set A and set C, where set B is not the same as set C. Since there are different sets involved, then the correspondences are also different.I don’t see how there can be two different one-to-one correspondences
Your logic assumes finite sets, not infinite sets. Since we are dealing with countably infinite sets here, then you logic does not apply.IF one to one correspondence means there are only pairs between the two sets. If that is the definition, the addition of even one number to one or the other set would make it larger.