I didn't get why you chose that clause in particular. I see now - it was just the nominated example. — TonesInDeepFreeze
(Disclaimer yes, apology and placation no.) — TonesInDeepFreeze
like saying that the score in a baseball game is tied -- without saying what the score is. — fishfry
"Cantor realized that [the set of natural numbers is 1-1 with the set of odd numbers]". — TonesInDeepFreeze
But there are infinite more natural numbers, just as with the reals. — Gregory
Is the point that there are far more infinities of reals than infinity of naturals vs the odd? — Gregory
I can imagine putting any two infinities one to one — Gregory
If you have an alternative theory, then state your axioms. — TonesInDeepFreeze
Impressionistic descriptions are fine for stoking creativity in mathematics and sometimes for making certain mathematical concepts intuitive. But they are not mathematical demonstrations. — TonesInDeepFreeze
I really want to know — Gregory
You really are not familiar with the proof? — TonesInDeepFreeze
I've asked people many times and they bring up the diagonal thing, although this just shows there are infinity more uncountable than countable and yes, however there are infinity many natural than odd. But you can biject with one and not the other? I'm not a jerk, just want some way I can understand what they are saying. It seems to me infinity is always just infinity at the end — Gregory
Incorrect: We should not use 'least' if we don't mean quantity.
It is typical of cranks unfamiliar with mathematical practice to think that the special mathematical senses of words most conform to their own sense of the words or even to everyday non-mathematical senses. The formal theories don't even have natural language words in them. Rather, they are purely symbolic. Natural language words are used conversationally and in writing so that we can more easily communicate and see concepts in our mind's eye. The words themselves are often suggestive of our intuitions and our conceptual motivations, but proofs in the formal theory cannot appeal to what the words suggest or connote. And for any word such as 'least' if a crank simply could not stomach using that word in the mathematical sense, then, if we were fabulously indulgent of the crank, we could say, "Fine, we'll say 'schmleast' instead. 'schmardinality' instead'. 'ploompty ket' instead of 'empty set' ... It would not affect the mathematics, as the structural relations among the words would remain, and the formal symbolism too. — TonesInDeepFreeze
df: K is a cardinal iff K is an ordinal and there is no ordinal j less than K such that there is a bijection between K and j.
There is no mention of 'cardinal' or 'cardinality' in the definiens. — TonesInDeepFreeze
You wouldn't call it "my" theory of relativity, or "my" theory of evolution, just because I happened to invoke those well-established scientific ideas in a conversation. — fishfry
It's a bit like saying that the score in a baseball game is tied -- without saying what the score is. Maybe that helps. — fishfry
If one thing is defined in terms of some other thing, the latter is logically prior. As is the case with cardinal numbers, which are defined as particular ordinal numbers. — fishfry
I'd agree that given some ordinal number, it's cardinally equivalent to some other sets. It doesn't "have a cardinality" yet because we haven't defined that. We've only established that a given ordinal is cardinally equivalent to some other set. — fishfry
Note per your earlier objection that by "least" I mean the ∈∈ relation, which well-orders any collection of ordinals. If you prefer "precedes everything else" instead of "least," just read it that way. — fishfry
No. Cardinal equivalence is logically prior to ordinals in the sense that every ordinal is cardinally equivalent to some other sets. At the very least, every ordinal is cardinally equivalent to itself.
When you use the word "cardinality" you are halfway between cardinal numbers and cardinal equivalence, so you confuse the issue. Better to say that cardinal equivalence is logically prior to ordinals; and that (in the modern formulation) ordinals are logically prior to cardinals. — fishfry
We don't need to suppose toward contradiction that there is a surjection. — TonesInDeepFreeze
With my original question I didn't think too hard on the point that if one did take a drop of water from an infinite ocean they would have no place to take it. And if I created infinite land next to an infinite ocean that might create even more questions. — TiredThinker
Or can ripples or motion itself exist in this ocean? — TiredThinker
I believe TheMadFool gave this example earlier, where we can start with the infinite set 1, 2, 3, 4, ..., then remove 1 to leave 2, 3, 4, ... What's left is still infinite, yet it's missing 1. That can happen too. Infinity is funny that way. — fishfry
OK Tones, explain to me then what "least" means in "the mathematical sense", if it is not a quantitative term. It can't be "purely symbolic" in the context we are discussing. For example, when fishfry stated von Neumann's definition of a cardinal as "the least ordinal having that cardinality", through what criteria would you determine "least", if not through reference to quantity? — Metaphysician Undercover
Here's another example. Look at your use of "less than". How is one ordinal "less than" another, without reference to quantity? — Metaphysician Undercover
Yes, in the context of the example we are discussing, I would. Unless you were quoting it word for word from another author, or explicitly attributing it to someone else, I would refer to it as your theory. I believe that is to be expected. Far too often, Einstein's theory, and Darwin's theory are misrepresented,. So instead of claiming that you are offering me 'Cantor's theory', it's much better that you acknowledge that you are offering me your own interpretation of 'Cantor's theory', which may have come through numerous secondary sources, unless you are providing me with quotes and references to the actual work. — Metaphysician Undercover
OK, so let's start with this then. In general we cannot determine that a game is tied without knowing the score. However, if we have some way of determining that the runs are equal, without counting them, and comparing, we might do that. Suppose one team scores first, then the other, and the scoring alternates back and forth, we'd know that every time the second team scores, the score would be tied, without counting any runs. Agree? Is this acceptable to you, as a representation of what you're saying? — Metaphysician Undercover
to many other sets, including itself. But when we clarify this terminology, your sophistic point evaporates.This means that cardinality is a property of all ordinals, [/qmote]
No no no no no. No. Every ordinal is cardinally equivalent — Metaphysician Undercover
it is an essential, and therefore defining feature of ordinals. — Metaphysician Undercover
So we have a sense of "cardinality" which is logically prior to ordinals, as inherent to all ordinals, — Metaphysician Undercover
and we also have a sense of "cardinal" number which is specific to a particular type of ordinal. — Metaphysician Undercover
Don't you see how this is becoming nonsensical? — Metaphysician Undercover
What you are saying is that it has a cardinality, — Metaphysician Undercover
because it is cardinally equivalent to other sets, but since we haven't determined its cardinality, it doesn't have a cardinal number. — Metaphysician Undercover
In essence, you are saying that it both has a cardinality, because it is cardinally equivalent, — Metaphysician Undercover
and it doesn't have a cardinality because it's cardinality hasn't been determined, or assigned a number. — Metaphysician Undercover
Let's look at the baseball analogy. We know that the score is tied, through the equivalence, so we know that there is a score to the game. We cannot say that because we haven't determined the score there is no score. Likewise, for any object, we cannot say that it has no weight, or no length, or none of any other measurement, just because no one has measured it. What sense does it make to say that it has no cardinal number just because we haven't determined it? — Metaphysician Undercover
Actually, this explains nothing to me. "Precedes" is a relative term. So you need to qualify it, in relation to something. "Precedes" in what manner? — Metaphysician Undercover
Yes, this demonstrates very well the problem I described above. Because the set has a "cardinal equivalence, — Metaphysician Undercover
it also necessarily has a cardinality, — Metaphysician Undercover
and a corresponding mathematical object which you call a cardinal number. — Metaphysician Undercover
Why do you think that you need to determine that object, the cardinal number, before that object exists as the object which it is assumed to be, the cardinal number? — Metaphysician Undercover
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