A one to one correspondence of E to N, with the conclusion 'there are as many even integers as there are integers'. — sandman
Not in direct opposition to this statement, but just a remark...Informally, and for so long as you and your readers understand the formality underlying the informality, yes. — tim wood
However, let's do something different. We take the same sets N and E. We know that N has the even numbers. So we pair the members of E with the even numbers in N. We can do that perfectly and with each member of E in bijection with the even number members of N. What now of the odd numbers in N? They have no matching counterpart in E.
Doesn't this mean N > E? — TheMadFool
↪fishfry
Do you agree that there exists at least one bijection from E to N?
— fishfry
So there was nothing wrong when Socrates defined humans as featherless bipeds and someone came along with a chicken plucked of all its feathers and declared "this is a human"? After all there was/is at least one human that fit the definition. — TheMadFool
I can only implore you to carefully re-read what I and others have written. Perhaps Cantor's beautiful ideas will come to you at some point. Perhaps not. — fishfry
[Direction is not a factor in forming a sequence.]I would just flip bits along the diagonal and have a sequence they didn't include on their list. — ee
[A well known mathematician, whose name escapes me, when asked to define mathematics replied "a manipulation of symbols". I was impressed by such a concise and accurate description.]One can think of it as a game with symbols. — Eee — ee
[Direction is not a factor in forming a sequence.] — sandman
[A well known mathematician, whose name escapes me, when asked to define mathematics replied "a manipulation of symbols". I was impressed by such a concise and accurate description.] — sandman
I really hope so too.Thanks. — TheMadFool
It's not such a bad association, although I do not know what you mean by it. I am under the impression that Cantor's major contribution wasn't inventing or discovering a concept called infinity - that was old news. What he did do was develop ideas of transfinite cardinals and ordinals along with an arithmetic of transfinite cardinal and ordinal numbers. That is, of different sized infinities, of which there are apparently a whole lot.He associated the truly infinite with GOD. He should have left it there. — sandman
Or as Charles Sanders Peirce aptly put it, mathematics is the science that draws necessary conclusions about purely hypothetical states of things.In this case there is no underlying reality. There is no "setness" that we are trying to formalize. Rather, sets are whatever satisfies the rules we're writing down. Before the rules are written down, there are no sets! — fishfry
Ironically, there is a countably infinite number of cardinals, only the smallest of which is itself countable.What he [Cantor] did do was develop ideas of transfinite cardinals and ordinals along with an arithmetic of transfinite cardinal and ordinal numbers. That is, of different sized infinities, of which there are apparently a whole lot. — tim wood
The real numbers are at the center of both practice and intuition. While you have focused on Cantor, the nature of the real numbers as theoretical entities is just as strange and philosophically questionable. Why aren't you railing against the 'superstition' that 2 has a square root? Have you ever seen it? It's a theoretical construction — ee
Or as Charles Sanders Peirce aptly put it, mathematics is the science that draws necessary conclusions about purely hypothetical states of things. — aletheist
Ironically, there is a countably infinite number of cardinals, only the smallest of which is itself countable. — aletheist
I wouldn’t label it ‘superstition’, but an abstraction, like point, line, circle, the continuum, etc., all mental constructs for purposes of measurement. They are practical conveniences — sandman
Here we have what I think is an inconsistency - the same rule (pairing elements of one set with elements of another) producing, depending on the way you do the pairing, different, actually contradictory, results. Math can't have contradictions can it? — TheMadFool
What you leave out, and what has apparently been left out, of all of this is that the sets have to first be well-ordered. Then the bijection is a two-way Hobson's choice: next rider, next horse. And you never run out of either riders or horses. The problem with irrationals, is that they cannot be put into a well-ordering. — tim wood
Here we have what I think is an inconsistency - the same rule (pairing elements of one set with elements of another) producing, depending on the way you do the pairing, different, actually contradictory, results. Math can't have contradictions can it? — TheMadFool
Nope. Google bijection.* The unit interval [0,1] can be bijected to the interval [0,2] but neither set is well-ordered in its usual order and both are uncountable sets full of irrationals (and some rationals too).
* Ok you say but at least in that case the obvious bijections preserve order. But how about bijecting [0,1] to the open unit interval (0,1)? Then there's a bijection but it does not preserve order. — fishfry
what is their usual order? What matters is that they can be well-ordered.* The naturals biject to the rationals and the rationals in their usual order are not well-ordered. — fishfry
Nope. Google bijection. — tim wood
* The naturals biject to the rationals and the rationals in their usual order are not well-ordered.
— fishfry
what is their usual order? What matters is that they can be well-ordered. — tim wood
st of your post, let's try this. Name any two real numbers that are next to each other in a well-ordering of the set of real numbers. — tim wood
Granted, given two such numbers as a set of two numbers, the elements of that set can be well-ordered, but the set in question is the set of reals. — tim wood
The real problem with all of this stuff is not that everyone else is wrong and you alone are correct - no danger of that. Nor even that you cannot handle it, because you can. The real difficulty with these ideas is just getting used to them. So in the words of my neighbor, a retired master sergeant, who so far has always made sense, suck it up, buttercup, and get used to them — tim wood
Nope. Google bijection. — tim wood
herefore [0,1] and [0,2] are of equal cardinality. — fdrake
I think you need to check your math privilege. — fdrake
I just don't understand the insult culture around here, especially involving technical matters. — fishfry
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