For me it can be difficult to tell whether something's technical or not if I'm unfamiliar with it. — fdrake
come to this forum in the hopes that I DON'T have to bean Internet jerk just to hold my own in a conversation. — fishfry
I accept. I ought to because I've delivered a few and felt justified doing so, so I ought to take if I've earned. But I still have a problem with bijection in uncountable sets: how do you do it? If defined by equal cardinality, then my bad. But that doesn't answer my question as to how it's done.Hey tim wood go fuck yourself. — fishfry
Do you mean the integers? I accept that there is a "usual ordering" of the integers, but what would "the usual order" of the rationals look like? Or do you mean that any well-ordering is "the usual"?What is the usual order on the rationals? You aren't sure? Come on, man. The usual order is the usual order. — fishfry
Try this site:Yet, following the same principle - pairing one element to another of these two sets - we arrive at a situation where every even number is paired with an even number in the set of natural numbers with the odd numbers left unpaired. — TheMadFool
Hmm. Being called out is a good incentive to do some research and learn something. I find this among many sites that mention the problem. This one is brief and readable.Which can be well-ordered: definitely using the axiom of choice; and possibly even in the absence of choice. — fishfry
There is no contradiction. The definition of cardinal equivalence is that there exists at least one bijection between the two sets. I don't know why you have a psychological block against grokking that.
Guy robs a bank, gets caught. In the interrogation room the detective says, "Fred we know you're the bank robber." Fred says, "Oh you are wrong. Here is a list of all the banks in the state that I didn't rob. I even have a notarized statement to that effect from the manager of every single bank in the country that I did not rob."
Is Fred a bank robber? Yes of course. He robbed a bank! He robbed one single solitary bank and DIDN'T rob all the others. But he's a bank robber.
It's an existential quantification, "there exists," and not a universal one, "for all." Someone is a bank robber if they ever robbed a bank, even if there are many banks they didn't rob. Two sets are cardinally equivalent if there is a bijection between them, even if -- as must ALWAYS be the case -- there are maps between them that are not bijections.
Someone murders someone, they're a murderer. No use parading before the jury the seven billion human beings they DIDN"T murder. That lady cop in Dallas a few months ago who shot a guy sitting in his living room eating ice cream. She was convicted of murder. She's in prison as we speak, ten years if I recall. No use trying to point to all of her neighbors who she didn't kill. She killed one guy. That makes her a murderer in the eyes of the law.
Why is this simple point troubling you? If you're on the jury do you say, "Well, the prosecutor showed that she murdered someone. But she didn't murder EVERYONE." You find her not guilty on that basis? Of course not! Right?
Even Hitler didn't murder EVERYONE. You think he got a bad rap? LOL. — fishfry
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. (I.e., whenever you put two net to each other, you can then always put one in between - actually, a whole infinity of numbers in between.) — tim wood
In terms of your example of even onto the even integers, and evens onto all integers, the latter, because the odd integers aren't matched, isn't a bijection. — tim wood
I mean by well-ordered a set (of numbers) ordered in such a way that given a starting value, say, on the left, one could move to the right step-by-step and enumerate/list/count all the elements without missing any. The natural numbers, ordered on ≤, would be such an ordering.What do you mean well ordered? Kindly explain. — TheMadFool
https://math.berkeley.edu/~kpmann/Well-ordering.pdf
"So what do we believe today? First of all, it has been shown that if you want to believe the well-ordering theorem, then it must be taken as an axiom." The author goes on to observe that it's a pretty good axiom, and most accept it. He also notes in a footnote that "Moreover, it has also been proven that it is impossible to write down an explicit well-ordering for the set of real numbers."
I mean by well-ordered — tim wood
I mean by well-ordered a set (of numbers) ordered in such a way that given a starting value, say, on the left, one could move to the right step-by-step and enumerate/list/count all the elements without missing any. The natural numbers, ordered on ≤, would be such an ordering. — tim wood
I just don't understand the insult culture around here. Over the past couple of years I've had to take extended breaks from this forum because someone started piling on personal insults at me over technical matters on which they happened to be flat out wrong. Not because I can't snap back; but because I'm perfectly capable of snapping back, and that's not what I'm here for. I'd suggest to members that whenever they throw an insult in lieu of a fact, perhaps they should consider whether they've got any facts. — fishfry
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. — TheMadFool
How about learning actual math? Set theory in this case. It's easy in our time. Just google it. I'll even give a link here: Well-order. Or here: Well ordered set or Well-ordering principle.What do you mean well ordered? Kindly explain. — TheMadFool
The problem with your argument is that the described function is not bijective. Instead, you have constructed an injection from the set of even numbers to the set of natural numbers. From this, you are only permitted to conclude that the set of even numbers has at least as many elements as the set of natural numbers. If you combine this with an injection from the set of natural numbers to the set of even numbers, then you can conclude that a bijection exists. Because these sets are countable, this bijection is constructible (and is given by the standard map from the natural numbers to the even numbers). — quickly
Showing that you don't quite understand what others are talking about isn't a great argument. — ssu
What if the counting will go on forever - not a trivial problem. Answer: you define counting and establish some rules for it, then you count and if the count is even at every point, then the two things are same size. — tim wood
One-to-one correspondence gives nonsense results. — Devans99
Did we define "exist"? Seems like it might be a good idea.Infinity does not exist — Devans99
I am, about some things, like 2+2=4. But for you I'll reopen a very open-minded offer I made to you earlier. I give you one dollar bills, and for each one you give me a five dollar bill. Can't get much more open-minded than that, or is that too much for you?- You are very closed minded. — Devans99
Did we define "exist"? Seems like it might be a good idea. — tim wood
I am, about some things, like 2+2=4. But for you I'll reopen a very open-minded offer I made to you earlier. I give you one dollar bills, and for each one you give me a five dollar bill. Can't get much more open-minded than that, or is that too much for you? — tim wood
Yes. Let's be clear. You said Infinity doesn't exist, now that it does. I agree that it does, as an idea. And I'm inclined to agree that it cannot exist in reality - although reality itself does present some challenges to that (e.g., a ruler, and by extension the rest of reality).Does infinity have existence beyond our minds? All sorts of things like fairies, square circles, can exist in our minds — Devans99
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