So what would your replacement be? — Banno
I did not think this was an appropriate context in which to mention the one-point compactification of the real line. — fishfry
Not a major point in context of this thread; but it is a technicality that should not be deined. — TonesInDeepFreeze
They told me not to argue with the teacher. — TonesInDeepFreeze
You must have driven your teachers crazy. — fishfry
Merely, I added stated the qualification, and said that it should not be denied, while not meaning to imply that you personally denied it. — TonesInDeepFreeze
Throughout the thread, it seems to me that regularly take technical and heuristic disagreements, corrections, and even mere technical qualifications as attempts to undermine you personally. — TonesInDeepFreeze
It's in your obfuscatory and unnecessarily argumentative mind that anyone denied it. — fishfry
Rudin, the number one, main, classic real analysis text, that defines the extended reals exactly as I did — fishfry
Yes, in many (probably most or even just about all) writings, the points of infinity are just arbitrary points, and they are not specified to be any particular mathematical objects. — TonesInDeepFreeze
You are wrong on the pedagogy AND wrong on the facts. — fishfry
I didn't say that anyone denied it. I said it shouldn't be denied. — TonesInDeepFreeze
Your pickiness with everything I write annoys me. — fishfry
Especially because half the time you're actually wrong on the facts. — fishfry
I didn't say that anyone denied it. I said it shouldn't be denied.
— TonesInDeepFreeze
I just laughed, man. — fishfry
I think we're two of a kind. — fishfry
Peace — fishfry
So yes, cardinality is already inherent within the ordinals. Each ordinal has a cardinality. I — fishfry
Not at all. Not "more or less," but "prior in the order," if you prefer more accurate verbiage.
You insist on conflating order with quantity, and that's an elementary conceptual error. In an order relation x < y, it means that x precedes y in the order. x is not "smaller than" y in a quantitative sense. I can't do anything about your refusal to recognize the distinction between quantity and order. — fishfry
I wouldn't. I would say they are different predicates of the form: x is infinite & Rx. — TonesInDeepFreeze
If someone took a single drop of water of finite size from an infinite ocean would it actually be taking from the ocean? — TiredThinker
Well then it's incorrect to say that ordinality is logically prior to cardinality. — Metaphysician Undercover
ordinals are logically prior to cardinals — fishfry
If there is already cardinality inherent within ordinality then the closest you can get is to say that they are logically codependent. But if order is based in quantity, then cardinality is logically prior. — Metaphysician Undercover
"Least", lesser, and more, are all quantitative terms. — Metaphysician Undercover
Cardinality is inherent. — fishfry
In other words, you agree that it's incorrect to say that ordinals are logically prior to cardinals. — Metaphysician Undercover
despite my lack of understanding of your "bijective equivalence" — Metaphysician Undercover
I would make clear that 'infinity' and 'infinite' are not be be conflated — TonesInDeepFreeze
If someone took a single drop of water of finite size from an infinite ocean would it actually be taking from the ocean? Would the ocean replace that exact drop immediately upon it being taken or would it simply never matter? — TiredThinker
ordinals are logically prior to cardinals, in the modern formulation. — fishfry
the equivalence class of all sets having that cardinality. The problem was that this was a proper class and not a set — fishfry
two sets having the same cardinality -- meaning that there is a bijection between them -- and assigning them a cardinal -- a specific mathematical object that can represent their cardinality. — fishfry
After von Neumann, we identified a cardinal with the least ordinal of all the ordinals having that cardinality — fishfry
Any nonempty collection of ordinals always has a least member — fishfry
conflating order with quantity [is] an elementary conceptual error. In an order relation x < y, it means that x precedes y in the order. x is not "smaller than" y in a quantitative sense. — fishfry
cardinality, which is an equivalence relation based on bijection — fishfry
Two sets may have the same cardinality, without there being any notion of ordinal at all — fishfry
I am not suggesting that my comments supplant yours. — TonesInDeepFreeze
I am not suggesting that my comments supplant yours.
— TonesInDeepFreeze
LOL — fishfry
But you say it's "my" bijective equivalence as if this is some personal theory I'm promoting on this site. On the contrary, it's established math. You reject it. I can't talk you out of that. — fishfry
Two sets are bijectively equivalent if there is a bijection between them. In that case we say they have the same cardinality. We can do that without defining a cardinal number. That's the point. The concept of cardinality can be defined even without defining what a cardinal number is. — fishfry
LOL
— fishfry
I don't know what your point is there. — TonesInDeepFreeze
No, I think you misinterpret this. I say it's "your" bijective equivalence, because you are the one proposing it, not I. — Metaphysician Undercover
So "yours" is in relation to "mine", and anyone else who supports your proposition (even if you characterize it as "established math") is irrelevant. If you wish to support your proposition with an appeal to authority that's your prerogative. In philosophy, the fact that something is "established" is not adequate as justification. — Metaphysician Undercover
This is what I do not understand. Tell me if this is correct. Through your bijection, you can determine cardinality. — Metaphysician Undercover
But are you saying that you do this without using cardinal numbers? What is cardinality without any cardinal numbers? — Metaphysician Undercover
What I think is that you misunderstand what "logically prior" means. Here's an example. We define "human being" with reference to "mammal", and we define "mammal" with reference to "animal". Accordingly, "animal" is a condition which is required for "mammal" and is therefore logically prior. Also, "mammal" is logically prior to "animal". You can see that as we move to the broader and broader categories the terms are vaguer and less well defined, as would happen if we define "animal" with "alive", and "alive" with "being". In general, the less well defined is logically prior. — Metaphysician Undercover
Now let me see if I understand the relation between what is meant by "cardinality" and "cardinal number". Tell me if this is wrong. An ordinal number necessarily has a cardinality, so cardinality is logically prior to ordinal numbers. — Metaphysician Undercover
And to create a cardinal number requires a bijection with ordinals, so ordinals are logically prior to cardinal numbers. — Metaphysician Undercover
Where I have a problem is with the cardinality which is logically prior to the ordinal numbers. — Metaphysician Undercover
It cannot have numerical existence, because it is prior to ordinal numbers. Can you explain to me what type of existence this cardinality has, which has no numerical existence, yet is a logical constitutive of an ordinal number. — Metaphysician Undercover
Well, at least thank you for not saying 'thank you'. — TonesInDeepFreeze
I said LOL because I was amused/charmed — fishfry
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