For the set of natural numbers and the set of odd numbers, 2n-1 is the relationship that yields a one-to-one correspondence between them. For other sets, a different formula (not equation – an equation has an equals sign in it, as the name indicates) would serve the same purpose.The 2n-1 equation is not relevant. You can put the one to one correspondence which you believe in together without it, no? So it adds nothing. Yet I disagree with the one to one correspondence at any level. 1 2 3 4 5 6 7 8 9 10 do NOT correspond to 1 3 5 7 9. That’s reality
Do you honestly think that the best explanation of the situation here is that your “careful concentration” has uncovered an error or deception that the vast majority of people who devote their lives to studying this topic have never noticed? If it were that easy a mistake to uncover, doesn’t it stand to reason that mathematicians would long since have discarded this “irrational thesis” and moved on to something else?Gamow and Cantor both used sleight of hand to convince people of their irrational thesis. But careful concentration shows there error.
Drs Dunning and Kruger have a great deal to say on this point, I think.Do you honestly think that the best explanation of the situation here is that your “careful concentration” has uncovered an error or deception that the vast majority of people who devote their lives to studying this topic have never noticed? If it were that easy a mistake to uncover, doesn’t it stand to reason that mathematicians would long since have discarded this “irrational thesis” and moved on to something else?
When I said that the 2n-1 relation was arbitrary, I didn’t mean that 2 times 3 minus 1 wasn’t 5, but that its function in your argument was faulty. Just because lots of people believe it don’t make it true, nor that other things they believe are false. Once people were considered a fool to believe that geocentrism was false. 2n-1 can’t crunch 1 3 5 7 9 to line up with 1 2 3 4 5 6 7 8 9 10 and that is the essence of my argument. 1 2 3 4 5 6 7 8 9 10 is larger, and it **will continue to be **as you multiply 1 3 5 7 9 and 1 2 3 4 5 6 7 8 9 10 to infinity.For the set of natural numbers and the set of odd numbers, 2n-1 is the relationship that yields a one-to-one correspondence between them. For other sets, a different formula (not equation – an equation has an equals sign in it, as the name indicates) would serve the same purpose.
And I have agreed numerous times thar {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} cannot be placed into one-to-one correspondence with {1, 3, 5, 7, 9}. No one has claimed that it can, since one set has twice as many members as the other. On the other hand, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} does correspond one-to-one with {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}. You are hung up on stopping at 10 both times, when the question we are interested in has nothing to do with that but is instead all about how many members there are.
Usagi
I can see where your conceptual error is. I wish I had a blackboard, that would make this so much easier. I’ll try here.When I said that the 2n-1 relation was arbitrary, I didn’t mean that 2 times 3 minus 1 wasn’t 5, but that its function in your argument was faulty. Just because lots of people believe it don’t make it true, nor that other things they believe are false. Once people were considered a fool to believe that geocentrism was false. 2n-1 can’t crunch 1 3 5 7 9 to line up with 1 2 3 4 5 6 7 8 9 10 and that is the essence of my argument. 1 2 3 4 5 6 7 8 9 10 is larger, and it **will continue to be **as you multiply 1 3 5 7 9 and 1 2 3 4 5 6 7 8 9 10 to infinity.
No. We have shown that the infinity of the reals does not equal the infinity of the naturals.Also, your method could be used to make ANY two infinities seem equal.
I don’t disagree that that the infinity of reals are greater, but the method used by Usagi would prove otherwise if it was valid. Read my last post, and then you see he is trying to count to infinityNo. We have shown that the infinity of the reals does not equal the infinity of the naturals.
No. A slight variation on the method used by Usagi proves the opposite.I don’t disagree that that the infinity of reals are greater, but the method used by Usagi would prove otherwise if it was valid. Read my last post, and then you see he is trying to count to infinity
I’m not forgetting 11-15 in the first column. We just didn’t show them in the sample. Remember - both of these sets are unbounded. I just showed the first 10 elements of each set… Above, you’re showing the first 10 elements of the first set, and the first 5 elements of the second set. Your problem is that you keep stopping - you’re using a bound set and trying to apply the pattern you see to the unbounded, infinite sets.But the method is faulty for two reasons. You have
1 1
2
3 3
4
5 5
6
7 7
8
9 9
10
You method puts the 3 to 2, the 5 to 3, 7 to four, ect., but** it is forgetting 11-15 in the first column**. See my point?? You have 1 2 3 4 5 6 7 8 9 10 being greater than 1 3 5 7 9. When you multiply these sets to infinity, you have an equal number of such sets, but there are more units in the first, because 1 3 5 7 9 don’t have a one to one correspondence to 1 2 3 4 5 6 7 8 9 10
Also, your method could be used to make ANY two infinities seem equal. Since you believe there is such a thing as unequal infinities, this method is not right
Odd numbers all the way to infinity are a part of the whole natural numbers and are, thusly, all the way up to infinityThis is just a general observation, but perhaps it will help:
People often have trouble with the notion of different sizes of infinity because the commonly used examples involve a set being compared to one of its subsets, such as the integers being placed in correspondence with the odd numbers. Such examples are counterintuitive because we tend to imagine the subset “lying inside” the larger set, so it seems absurd for the sets to be of equal size. However, correspondence is meant to offer a way of comparing two sets that may be entirely unrelated; how one may or may not be included in the other is completely disregarded.
So if we use correspondence to compare two sets, one of which contains the integers and the other contains, say, all the words representing only the odd integers (just the words, not the integers themselves), then it suddenly becomes much more intuitive. After all, why should the type of objects in the sets impact their size?
(1) If the set of all whole numbers and the set of whole odd numbers are of different sizes, then they will not be able to be put into a one-to-one correspondence.1 2 3 4 5 6 7 8 9 10 is greater than 1 3 5 7 9. When you multiply these sets to infinity, you have an equal number of such sets, but there are more units in the first. Please explain how any principle that forces the second set to be equal to the first cannot be used to make any two infinities equal. From how it was explained in the book it referred to, there is no such principle: you can line up some ten numbers and say they are equal because they go, from that one to one correspondence of the first ten, then to infinity