I answered this in my most recent post to you. Given two ordinals, it's always the case that one is an element of the other or vice versa. — fishfry
This is NOT true of sets in general, but it IS true for ordinals, and that's what makes the construction work. — fishfry
No, as I'm pointing out to you. It's true that every ordinal is cardinally equivalent to itself, but that tells us nothing. You're trying to make a point based on obfuscating the distinction between cardinal numbers, on the one hand, and cardinal equivalence, on the other. — fishfry
Other way 'round. A cardinal number is defined as a particular ordinal, namely the least ordinal (in the sense of set membership) cardinally equivalent to a given set. — fishfry
Right. I can live with that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is. — fishfry
Cardinal equivalence is a relation between two sets. It's not something a set can have by itself. — fishfry
I see where you're going with this. Given a set, it has a cardinal number, which -- after we know what this means -- is its "cardinality." You want to claim that the set's cardinality is an inherent property. But no, actually it's a defined attribute. First we define a class of objects called the cardinal numbers; then every set is cardinally equivalent to exactly one of them. But before we defined what cardinal numbers were, we couldn't say that a set has a cardinal number. I suppose this is a subtle point, one I'll have to think about. — fishfry
But when you got up that morning, before you came to my party, you weren't a room 3 person or whatever. The assignment is made after you show up, according to a scheme I made up. Your room-ness is not an inherent part of you. — fishfry
Yes you are correct, it's cleaner to not use proof by contradiction. — fishfry
EyeX f(y) = S — TonesInDeepFreeze
Let f(y) = S — TonesInDeepFreeze
Is that a thing? — bongo fury
proving (still by contradiction) a stronger denial of surjectivity than mere failure of surjectivity. — bongo fury
EyeX f(y) = S
— TonesInDeepFreeze
which (I guess?) follows from
f(y) = S
— TonesInDeepFreeze — bongo fury
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 — Metaphysician Undercover
The proofs prove the exact same result - nothing more nothing less, — TonesInDeepFreeze
We don't need to suppose toward contradiction that there is a surjection. — TonesInDeepFreeze
That's backwards. — TonesInDeepFreeze
In that way, fishfry's RAA is deferred in my proof to later. — TonesInDeepFreeze
I didn't write EyeX f(y) = S as a separate line, since I didn't belabor certain obvious steps; it's not a fully formal proof. — TonesInDeepFreeze
RAA and modus tollens are basically the same. — TonesInDeepFreeze
suppose toward contradiction — TonesInDeepFreeze
RAA and modus tollens are basically the same. — TonesInDeepFreeze
For some mathematicians its a stylistic preference. — TonesInDeepFreeze
signpost[ing] — bongo fury
yours begins (read as a proof by contradiction) by denying a more specific claim of failure of surjectivity: the claim that such sets as, in particular, S will fail to be in the range of f. — bongo fury
In that way, fishfry's RAA is deferred in my proof to later.
— TonesInDeepFreeze
Yes. It's still a proof by contradiction, just not so upfront. — bongo fury
And it's still a proof by contradiction. — bongo fury
RAA and modus tollens are basically the same.
— TonesInDeepFreeze
Another reason not to expect an important contrast in your reworking. — bongo fury
And, of course, I wouldn't even think of denying the claim that S is not in the range of f. — TonesInDeepFreeze
And, of course, I wouldn't even think of denying the claim that S is not in the range of f.
— TonesInDeepFreeze
Except in a line properly signposted as RAA. — bongo fury
Yes you are correct, it's cleaner to not use proof by contradiction.
— fishfry
Is that a thing? Ok. — bongo fury
There's a double negative in what you're saying. — TonesInDeepFreeze
RAA premise would not need to deny ~P. Rather, in this case, the premise is P. — TonesInDeepFreeze
We don't need to suppose toward contradiction that there is a surjection. — TonesInDeepFreeze
Anyway, that's not the beginning of my proof. — TonesInDeepFreeze
Maybe they were undecided whether it was meant to be read as a proof by contradiction or not. — bongo fury
Not a big deal — fishfry
RAA premise would not need to deny ~P. Rather, in this case, the premise is P.
— TonesInDeepFreeze
Exactly, if for some reason you want to label the RAA line "P" rather "~P". In a line properly signposted as RAA, and in a discussion in which someone had bothered to say
We don't need to suppose toward contradiction that there is a surjection.
— TonesInDeepFreeze — bongo fury
it could make sense to display under that signpost (P or ~P depending on signposting preferences, or a form of words such as I chose so that the question didn't arise) the denial of what is to be shown. This denial will be the supposition toward a contradiction. What is to be shown is that S can't, without contradiction, be in the range of f. So the denial, the suitable RAA line, the supposition toward contradiction, if you or anyone did want to belabor the point, or understand the point about "not needing to suppose toward contradiction..." is indeed "S is in the range of f", and it might be interesting that this is taking the place of "f is surjective", in a proof by contradiction. — bongo fury
That incorrectly makes it appear that I said, "Incorrect: We should not use 'least' if we don't mean quantity." — TonesInDeepFreeze
OK, this makes more sense than what you told me in the other post, that one "precedes" the other. You are explaining that one is a part of the other, and the one that is the part is the lesser.. — Metaphysician Undercover
I assume that an ordinal is a type of set then. — Metaphysician Undercover
It consists of identifiable elements, or parts, some ordinals being subsets of others. — Metaphysician Undercover
My question now is, why would people refer to it as a "number"? — Metaphysician Undercover
Say for instance that "4" is used to signify an ordinal. What it signifies is a collection of elements, some lesser than others. — Metaphysician Undercover
By what principle is this group of elements united to be held as an object, a number? — Metaphysician Undercover
Do you know what I mean? — Metaphysician Undercover
A set has a definition, and it is by the defining terms that the sameness of the things in the set are classed together as "one", and this constitutes the unity of the set. — Metaphysician Undercover
In the case of the "ordinals", as a set, what defines the set, describing the sameness of the elements, allowing them to be classed together as a set? — Metaphysician Undercover
The issue, which you are not acknowledging is that "cardinal" has a completely different meaning, with ontologically significant ramifications, in your use of "cardinally equivalent" and "cardinal number". — Metaphysician Undercover
Let me explain with reference to your (I hope this is acceptable use of "your") hand/glove analogy. Let's take the hand and the glove as separate objects. Do you agree that there is an amount, or quantity, of fingers which each has, regardless of whether they have been counted? The claim that there is a quantity which each has, is attested by, or justified by, the fact that they are what you call "cardinally equivalent". So "cardinal" here, in the sense of "cardinally equivalent" refers to a quantity or amount which has not necessarily been determined. Suppose now, we determine the amount of fingers that the hand has, by applying a count. and we now have a "cardinal number" which represents the amount of fingers on each, the glove and the hand. In this sense "cardinal" refers to the amount, or quantity which has been determined by the process of counting. — Metaphysician Undercover
Do you agree with this characterization then? An ordinal is a type of set, and a cardinal is a type of ordinal. — Metaphysician Undercover
Logical priority is given to "set". — Metaphysician Undercover
So do you agree that a cardinal number is not an object, but a collection of objects, as a set? — Metaphysician Undercover
Or, do you have a defining principle whereby the collection itself can be named as an object, allowing that these sets can be understood as objects, called numbers? — Metaphysician Undercover
But this is an inaccurate representation. What you are saying, in the case of "cardinal numbers", is not "that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is", but that there is no "number" which corresponds with the amount of fingers in my glove, until it has been counted and judged. — Metaphysician Undercover
You can say, I know I have the same "amount" of fingers as my glove, but you cannot use "number" here, because you are insisting that the number which represents how many fingers there are, is only create by the count. — Metaphysician Undercover
Cardinal equivalence is a relation between two sets. It's not something a set can have by itself.
— fishfry
But you already said a set can be cardinally equivalent with itself. "If nothing else, every ordinal is cardinally equivalent to itself, so the point is made." — Metaphysician Undercover
Yes this exemplifies the ontological problem I referred to. — Metaphysician Undercover
Let's say "cardinality" is a definable attribute. Can we say that there is a corresponding amount, or quantity, which the thing (set) has, regardless of whether its cardinality has been determined? What can we call this, the quantity of elements which a thing (set) has, regardless of whether that quantity has been judged as a number, if not its "cardinality"? — Metaphysician Undercover
I see this as a very dangerously insecure, and uncertain approach, epistemically. See, your "scheme" is completely arbitrary. You may decide whatever property you please, as the principle for classification, and the "correctness" of your classification is a product simply of your judgement. In other words, however you group the people, is automatically the correct grouping.. The only reason why I am not a 3 person prior to going to the party is that your classification system has not been determined yet. If your system has been determined, then my position is already determined by my relationship to that system without the need for your judgement. It is your judgement which must be forced, by the principles of the system, to ensure a true classification. My correct positioning cannot be consequent on your judgement, because if you make a mistake and place me in the wrong room, according to your system, you need to be able to acknowledge this. and this is not the case if my positioning is solely dependent on your judgement. — Metaphysician Undercover
If you go the other way, as you are doing, then the position is determined by your subjective judgement alone, not by the true relation between the system of principles and the object to be judged. So if you make a mistake, and put me in the wrong room, because your measurement was wrong, I have no means to argue against you, because it is your judgement which puts me in group 3, not the relation between your system and me. — Metaphysician Undercover
inductive set well-ordered by ∈ — fishfry
I think you meant 'transitive set well ordered by ∈'. — TonesInDeepFreeze
Could someone rightfully say that 0, 1, and points are not in any sense sets? Or is there more too that? — Gregory
To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4,5, and so forth. And to be the number 4 is no more and no less than to be preceded by 3, 2, 1, and possibly 0, and to be followed by.... Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role -not by being a paradigm of any object which plays it, but by representing the relation that any third member of a progression bears to the rest of the progression.
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