• TonesInDeepFreeze
    1.8k
    So what would your replacement be?Banno

    I'd cobble together some of my remarks here with some other stuff. Whatever I did, I would make clear that 'infinity' and 'infinite' are not be be conflated, and explain that as I have here.
  • TonesInDeepFreeze
    1.8k
    I did not think this was an appropriate context in which to mention the one-point compactification of the real line.fishfry

    Doesn't have to be that. Could be just to choose any two mathematical objects that are not real numbers for +inf and -inf. For example, +inf = w ('w' for omega, standing for the set of natural numbers). Then the system is a certain specific mathematical object. Not a major point in context of this thread; but it is a technicality that should not be deined.
  • fishfry
    2.6k
    Not a major point in context of this thread; but it is a technicality that should not be deined.TonesInDeepFreeze

    Was it denied? Or simply omitted according to the common-sense principle of responding to a question at the level at which it was asked?

    I refer you to Rudin, Principles of Mathematical Analysis (pdf link), for decades the standard undergrad text for math major real analysis. On page 11 of the linked edition he says that the extended reals are the reals with two symbols adjoined.

    So you are wrong on the pedagogy AND wrong on the math. Not for the first time.

    They told me not to argue with the teacher.TonesInDeepFreeze

    Maybe I should call your parents.
  • TonesInDeepFreeze
    1.8k
    You must have driven your teachers crazy.fishfry

    I did. In fifth grade, the teacher showed a wall map of the acquisitions of U.S. territory. The map omitted the Gadsen Purchase and included it in the Mexican Cession. I said aloud in class that the map is wrong. She said it's not. I explained that it omits the Gadsen Purchase and that land was not obtained by the Mexican Cession. She said to be quiet. I told her that I would be quiet when she told the class that they should understand that the map is wrong. My parents were called. They told me not to argue with the teacher.
  • TonesInDeepFreeze
    1.8k


    I did not intend to imply that you personally denied it. But rather that it should not be denied.

    You said what the extended reals are. I noted a qualification.

    I did not say that you were personally amiss for not including that qualification, nor that you were not reasonable to deem it as too much detail for your purposes.

    Merely, I added stated the qualification, and said that it should not be denied, while not meaning to imply that you personally denied it.
  • fishfry
    2.6k
    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

    It's in your obfuscatory and unnecessarily argumentative mind that anyone denied it. You just made that up. Nobody denied anything. And I just gave you a link to Rudin, the number one classic real analysis text, that defines the extended reals exactly as I did, as the reals with two symbols adjoined having certain formal properties. You are wrong on the pedagogy AND wrong on the facts. My friend, if it's in Rudin, it's right. End of story.
  • TonesInDeepFreeze
    1.8k
    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.
  • fishfry
    2.6k
    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

    Your pickiness with everything I write annoys me. Especially because half the time you're actually wrong on the facts. I got bored of arguing with you in the other thread and I've achieved the same state of blissful detachment here.

    It may be that your pickiness annoys me because I have the same turn of mind, and we are always annoyed by those qualities in others that remind us of ourselves. That said, I've had enough for one day.
  • TonesInDeepFreeze
    1.8k
    It's in your obfuscatory and unnecessarily argumentative mind that anyone denied it.fishfry

    I didn't say that anyone denied it. I said it shouldn't be denied. And if that was not clear, I followed up in reply to say I did not intend to imply that it was denied.

    And there was no obfuscation.

    Rudin, the number one, main, classic real analysis text, that defines the extended reals exactly as I didfishfry

    Yes, I said in my original post:

    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

    I introduced my point to grant that. Then I said that we can also handle it another way. It's not unreasonable for me to say that.

    You are wrong on the pedagogy AND wrong on the facts.fishfry

    (1) I didn't make a claim about pedagogy. (2) t's not pedagogically inappropriate to add, essentially a footnote in this case, a certain technical qualification. (3) Posts don't need to restrict themselves to what is pedagogically best anyway.

    EDIT: And I'm not wrong on the substance.
  • fishfry
    2.6k
    I didn't say that anyone denied it. I said it shouldn't be denied.TonesInDeepFreeze

    I just laughed, man. I think we're two of a kind. Peace.
  • TonesInDeepFreeze
    1.8k
    Your pickiness with everything I write annoys me.fishfry

    It is not picky for me to say that you start your post with exaggeration: (1) There is a vast amount of what you write that I don't respond to, let alone with disagreement, correction, or qualification. (2) My points are not mere pickiness. That is only your own characterization.

    And whether something annoys you, you blow it way out of proportion, and often seemingly taking it to be improperly motivated against you.

    Especially because half the time you're actually wrong on the facts.fishfry

    If half the time I'm wrong, then the other half I'm right or at least neutral. Moreover, I have not been wrong half the time or anywhere remotely close.

    I started my post by saying that I basically agree with your post. Then I said I'm adding only a technical qualification. And I even said that most writers don't use the method I am mentioning but that others do and that it can be done. That's pretty damn mild.

    Then after one of your posts, I said that my point should not be denied. I didn't say that you denied it. My point was that no one should deny it. Then when you asked whether it had been denied and that you didn't mention it so as not to complicate things. Then I stated explicitly that I did not mean to imply that you denied it. And I will even add here that I don't blame you for thinking that I did mean to imply that you denied it. But after I've said it now about three times, you may correctly infer that I mean what I say when I say that I did not mean to imply that you denied it.
  • TonesInDeepFreeze
    1.8k
    I didn't say that anyone denied it. I said it shouldn't be denied.
    — TonesInDeepFreeze

    I just laughed, man.
    fishfry

    It is literally true that I did not say anyone denied it. And I haven't said that I blame you if you think I meant to imply that you denied it. And I don't blame you if you think that I meant to imply that you denied it. And above (though cross-posted) I say again that I did not mean to imply that you denied it.

    I think we're two of a kind.fishfry

    Yes, you are the rational, logical, reasonable, accommodating, conciliatory, factually correct, patient and pedagogically sagacious one. You are the Gallant of The Philosophy Forum. And I am the irrational, illogical, unreasonable, unaccommodating, non-conciliatory, factually incorrect, impatient, and pedagogically unwise one. I am the Goofus. And you're the better looking one too. The roles of hero and villain have been clearly scripted. Now for casting.

    Peacefishfry

    and Love, man.
  • Metaphysician Undercover
    10.6k
    So yes, cardinality is already inherent within the ordinals. Each ordinal has a cardinality. Ifishfry

    Well then it's incorrect to say that ordinality is logically prior to cardinality. 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.

    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

    "Least", lesser, and more, are all quantitative terms. So as long as you are using "least" to define your order, it is actually you who is conflating quantity with order. If you want a distinct order, which is not quantitative, you need something like "before and after", or "first and second". But first and second is a completely different conception from less and more, and would not be described by "least".

    If you want to emphasize a difference between quantity and order you need to quit using quantitative words like "least", when you are talking about order. However, I should remind you, that "least" is the term you used for Von Neumann's definition. If Von Neumann used the quantitative word "least", in his definition, then I think it is just a faulty interpretation of yours, which makes you insist on distancing quantity from order. In the reality of mathematical practise, order is defined by cardinality, not by anything like "first and second". So cardinality is held to be logically prior, regardless of what you claim.
  • Bradaction
    72
    I wouldn't. I would say they are different predicates of the form: x is infinite & Rx.TonesInDeepFreeze

    Ah Thank you!
  • Hanover
    9.1k
    If someone took a single drop of water of finite size from an infinite ocean would it actually be taking from the ocean?TiredThinker

    There aren't any infinite oceans.
  • fishfry
    2.6k
    Well then it's incorrect to say that ordinality is logically prior to cardinality.Metaphysician Undercover

    You're absolutely right. It would be incorrect to say that, because it's not true. I was pretty sure that I HADN'T said that, and I went back to page 2 of this thread and found what I actually said:

    ordinals are logically prior to cardinalsfishfry

    As you see, I said that ordinals are logically prior to cardinals. That's because cardinal numbers ARE particular ordinals. That's correct. That's what I said.

    Now I don't think your misquote of me was deliberately disingenuous. Rather, I think you don't have the mathematical sophistication to follow this conversation at all. Because in my previous post to you, I already explained the distinction between cardinality, which is an equivalence relation based on bijection; and cardinal numbers, which are particular ordinals. I see that went right over your head, leading to you inaccurately quote me based on your ignorance about what I already explained to you previously.

    And frankly I'm not going to get into it with you about this stuff. Go read my long article on the transfinite ordinals, or read the relevant Wiki page, or read a book on set theory. I can't argue with you about established, universally-accepted math. Unless you want to tell me what you think you know that John von Neumann didn't.


    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

    You lack the understanding to even know what you're saying. Again: Cardinality is inherent. How you define a cardinal number isn't. That you don't understand the distinction shows that you need to do a little homework on your own before you can credibly engage on this topic. There is no philosophical point involved. Two sets may have the same cardinality, without there being any notion of ordinal at all. But cardinal numbers are defined as particular ordinals. Cardinal numbers are subtly different than cardinality. I explained this to you previously, you either didn't read it or didn't understand it (not an exclusive or) and went ahead and deliberately misquoted me. It's tedious.

    "Least", lesser, and more, are all quantitative terms.Metaphysician Undercover

    Yeah yeah. I can't help you out. You should make an honest attempt to learn this material. I have made mighty efforts to explain basic order theory to you. You don't want to hear it. I am under no obligation to get into yet another conversation about this. I'm going to take my cue from John von Neumann here.
  • Metaphysician Undercover
    10.6k
    Cardinality is inherent.fishfry

    In other words, you agree that it's incorrect to say that ordinals are logically prior to cardinals. That is, unless you are just trying to hide a vicious circle by saying that a cardinal number is defined by its ordinality, and ordinality is defined by cardinality. But in the case of a vicious circle of two logically codependent things, it is still incorrect to say that one is logically prior to the other. So despite my lack of understanding of your "bijective equivalence", it is still you who is mistaken.
  • fishfry
    2.6k
    In other words, you agree that it's incorrect to say that ordinals are logically prior to cardinals.Metaphysician Undercover

    No, I said ordinals are logically prior to cardinals, in the modern von Neumann interpretation. I explained this several times. It's not right for you to hijack yet another thread by pointlessly trolling me like this.

    despite my lack of understanding of your "bijective equivalence"Metaphysician Undercover

    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.

    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.
  • Possibility
    2.7k
    I would make clear that 'infinity' and 'infinite' are not be be conflatedTonesInDeepFreeze

    Yep.

    Well, this is all fascinating on a mathematical level :yawn:, but the question in the OP was:

    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

    1. You can’t actually take from an infinite ocean, because there isn’t actually an infinite ocean to begin with. An ocean might appear infinite, but given that you are not an aspect of the ocean (and that you can remove a drop) renders any ocean you can speak of potentially finite (ie. at the very least it ends where you begin).

    2. An ocean is not a static, measurable object but an event - an ongoing process of evaporation and precipitation - and so is indeterminately quantifiable in terms of finite drops of water. Once you remove a single drop and recount, the quantity of finite drops of water in said ocean will have changed anyway, so there’s no way you could tell if you’d made any difference at all, even if you could immediately count all the drops. This is quantum physics.

    3. It could matter to the organisms living inside the drop that was removed, though. I couldn’t really say.

    There. You can go back to talking about infinities now...
  • jgill
    2.6k
    In complex analysis one speaks of all lines in the complex plane extending from the origin (0+0i) to "the point at infinity" - which makes little sense other than when one looks at the one-to-one correspondence between points in the plane and points on the Riemann sphere. The north pole "corresponds" to the hypothetical "point at infinity", which seems to exist in all directions, barely out of grasp.

    There are probably modern complex variables people who have more sophisticated ideas about complex infinity. Directed infinity is one such notion, in which the line has a particular slope (or the point at infinity has an argument). And then there are the various geometries and ideal points, manifolds, etc.

    So, there's quite a bit more about infinities or infinite than what is found in set theory. As I have mentioned, when I play with dynamical systems, infinite means unbounded in whatever context it appears and I rarely speak of the point at infinity :cool:
  • TonesInDeepFreeze
    1.8k
    fishfry's posts in this thread about ordinals have been generous and instructive. His posts deserve not to mangled, misconstrued, or strawmaned.

    I do not presume to speak for fishfry, but I would like to state some points, and add some points, in my own words too.

    fishfry:

    On matters of logic and mathematics, any divergence I take from you or qualification I mention is not intended as a criticism of you personally. (I am not suggesting that you have claimed or not claimed that I have intended my remarks on logic and mathematics to be personal criticisms of you.) My personal criticisms have only been about the dialogue itself.

    I am not suggesting that you would be remiss if you didn't adopt my formulations and definitions.

    I am not suggesting that my sometimes more detailed formulations supplant your sometimes less formal explanations (though in some instances I think your formulations are not correct).

    When I simply add detail or additional commentary, I am not suggesting that your original remarks are thereby incorrect.

    I am not suggesting that you would be remiss by not reading or not replying to anything I write (except rebuttals that are defenses against your incorrect criticisms of my claims).

    When I state something that you have already stated, I am not suggesting that you had stated to the contrary nor that you hadn't already stated it.

    I am not suggesting that you would be remiss not to the include the details I include.

    I am not suggesting that threads such this one require the detail that I include.

    I am not claiming that my formulations are pedagogically superior, as my intentions are not purely, or even primarily, pedagogical.

    I am not suggesting that my comments supplant yours.

    I am not suggesting that posting should be expected to keep a level of precision as we may expect in professional publication.

    If I misconstrue a poster, then it is unintentional. I do not intend to cause a strawman. If I have misconstrued a person, then they can let me know. But they would be incorrect if they claimed my error was intentional or that I had intentionally set up a strawman. (And I am not suggesting that you have or have not claimed that I have set up a strawman in previous threads.)

    I try to write mostly at face value. But in any communication it is often not clear whether a person meant to imply more than they literally said or not. If I have seemed to imply something that I did not literally write, then anyone can ask me whether I meant to imply it or not.

    /

    I am posting because I like talking about mathematical logic.

    I like expressing the concepts and explaining them.

    Sometimes my explanations are not understandable for people who are not familiar with mathematical logic, and in that case, I still enjoy having explanations and formulations available possibly for posters to revisit or even I enjoy just the fact that my remarks are on the record.

    I enjoy making formulations that are as rigorous as feasible in the confines of a thread.

    Posting sharpens my knowledge of the subject and improves my skill in composing formulations.

    When I post corrections to other posters, I find some small satisfaction in seeing that the correction is available to be read.

    I hope that some readers might benefit from my posts.

    I believe many of my formulations do provide insight and rigor and at least examples showing that rigorous formulations of certain notions exist.

    Sometimes I enjoy the interaction with posters.

    I enjoy reading some posters.

    I benefit from any true corrections or suggestions presented to my own posts.

    And with cranks, I find entertainment and satisfaction in providing counter to them.

    /

    ordinals are logically prior to cardinals, in the modern formulation.fishfry

    That is correct and it is important. It points to the fact that the set theoretic treatment of ordinals and cardinals is rigorous as it proceeds only step-by-step through the theorems and definitions.

    the equivalence class of all sets having that cardinality. The problem was that this was a proper class and not a setfishfry

    That is correct and it is important. For every S, there is the equivalence class of all sets that have a bijection with S. But that equivalence class is a proper set. So for a rigorous set theory, without proper classes, another definition of the cardinality operator needed to be devised. The numeration theorem ("for every S there is an ordinal T such that there is a bijection between S and T") is a theorem of ZFC, and it allows us to define card(S) = the unique ordinal k such that k has a bijection with S and such that no ordinal that is a member of k has a bijection with S.

    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

    That is correct and it is important. A set theoretic operator (such as 'the cardinality of') can be defined only by first showing that for every S, there exists a unique T such that T has a [fill in a certain property here]. For example, with the operator 'card' (meaning 'the cardinality of') we first prove:

    For every S, there is a unique T such that T is an ordinal and T has a bijection with S and no member of T has a bijection with S.

    card(S) = the unique T such that T is an ordinal and T has a bijection with S and no member of T has a bijection with S.

    After von Neumann, we identified a cardinal with the least ordinal of all the ordinals having that cardinalityfishfry

    I see the main point there, but the formulation is not clear to me. I suggest this sequence:

    df: k is ord-less-than j <-> k e j

    When context is clear, we just say "k is less than j" or "k < j" or "k e j".

    df: k is ord-least in S <-> (k e S & ~Ej(j e S & j ek))

    When context is clear, we just say "k is least in S".

    df: k is the least ordinal such that P <-> (k is an ordinal & Pk & ~Ej(Pj & j ek))

    df: the cardinality of S = the unique T such that T is the least ordinal having a bijection with S

    So the cardinality of S is card(S).

    df: S is a cardinal <-> (S is an ordinal and there is no ordinal T less than S such that there is a bijection between S and T)

    Any nonempty collection of ordinals always has a least memberfishfry

    That is correct and important. No clear understanding of ordinals and cardinals can be had without it.

    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

    That is correct and important especially in context of being presented in this thread with misconceptions about this. The less than relation on ordinals is simply the membership relation. That is, membership ALONE is the basis for the less than relation on ordinals. But a general quantitative relation for sets is formulated with cardinals that are based both on bijection and instantiated to specific sets with regard to the ordinal less than relation.

    cardinality, which is an equivalence relation based on bijectionfishfry

    I see what you intend, but to be precise, a relation is a set of tuples. But cardinality is not a set of tuples. The equivalence relation is among the cardinalities but is not the cardinalities themselves.

    Two sets may have the same cardinality, without there being any notion of ordinal at allfishfry

    I agree with the intent of that and I think perhaps some authors say things like that in a sense that does not require ordinals, but I find it not quite right.

    There are two different notions:

    (1) 'the cardinality of S' to mean the least ordinal k such that there is a bijection between S and k.

    and

    (2) 'S and T have the same cardinality' to mean there is a bijection between S and T

    In (1) 'cardinality of' is a 1-place operation.

    In (2) 'same cardinality' is a 2-place predicate.

    That's okay, except:

    (2) S and T have the same cardinality iff there is a bijection between S and T.

    In that sense, we don't need to rely on ordinals.

    (3) S and T have the same cardinality iff card(S) = card(T).

    In that sense, we do rely on ordinals.

    Which of (2) or (3) is the definition of 'same cardinality'? We wouldn't know unless the author told us, and whether the definition relies on ordinals would depend on what s/he told us the definition is.

    And (3) is better than (2) in the sense that (3) uses 'cardinality' compositionally from (1) while (2) takes 'cardinality' noncompositionally. And (3) better fits the usage such as: "What is the cardinality of S? It's the least ordinal k such that there is a bijection between S and k. Okay, so S and T have the same cardinality iff the cardinality of S is the cardinality of T."

    So I would just define:

    S and T have the same cardinality iff card(S) = card(T).

    Then take (2) as a theorem not a definition.

    And if we want to leave out ordinals, then just say "There is a bijection between S and T".

    /

    To clean up two misconceptions that have been expressed in responses to fishfry:

    Incorrect: The notion of ordinals presupposes the notion of cardinality.

    A definition of 'is an ordinal' does not refer to 'cardinal' nor 'cardinality', and it doesn't even refer to 'bijection'. That is a plain fact that can be verified by looking at any textbook on set theory.

    A definition of 'the order type of' does not refer to 'cardinality nor 'cardinality'. That is a plain fact that can be verified by looking at any textbook on set theory.. (The term 'ordinality' has been used. I am not familiar with it. Perhaps 'the ordinality of' means 'the order type of'?)

    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.
  • fishfry
    2.6k
    I am not suggesting that my comments supplant yours.TonesInDeepFreeze

    LOL I think you made your point. It's all good. Maybe you can straighten out @Metaphysician Undercover :-)
  • TonesInDeepFreeze
    1.8k
    I am not suggesting that my comments supplant yours.
    — TonesInDeepFreeze

    LOL
    fishfry

    I don't know what your point is there.
  • Metaphysician Undercover
    10.6k
    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

    No, I think you misinterpret this. I say it's "your" bijective equivalence, because you are the one proposing it, not I. 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.

    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

    This is what I do not understand. Tell me if this is correct. Through your bijection, you can determine cardinality. But are you saying that you do this without using cardinal numbers? What is cardinality without any cardinal numbers?

    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.

    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. And to create a cardinal number requires a bijection with ordinals, so ordinals are logically prior to cardinal numbers.

    Where I have a problem is with the cardinality which is logically prior to the ordinal numbers. 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.
  • TonesInDeepFreeze
    1.8k
    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.

    df: the cardinality of S = the least ordinal k such that there is a bijection between S and k.

    There is no mention of 'cardinal' or 'cardinality' in the definiens.

    /

    I don't use the term 'logically prior', but in context it probably would be fully explicated by induction on terms. Basically that a term T is prior to term Y iff the definiens in the definition of T does not depend on Y but the definiens in the definition of Y does depend on T. What requires induction on terms is the notion of 'depends'.

    In that sense 'ordinal' is prior to 'cardinal'
  • fishfry
    2.6k
    LOL
    — fishfry

    I don't know what your point is there.
    TonesInDeepFreeze


    When you do, you will be enlightened. :-)
  • fishfry
    2.6k
    No, I think you misinterpret this. I say it's "your" bijective equivalence, because you are the one proposing it, not I.Metaphysician Undercover

    I'm not proposing it, I'm reporting it from Cantor's work in the 1870's. 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. Minor semantic point though so let's move on.

    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

    So if I name-drop the theory of relativity or the theory of evolution I have to provide evidence? And if I don't I'm merely appealing to authority? Not an auspicious start to a post that actually did get better, so never mind this digression as to who gets credit for the idea of cardinal equivalence, and what is my burden of proof for simply mentioning that Cantor thought of it 150 years ago.

    This is what I do not understand. Tell me if this is correct. Through your bijection, you can determine cardinality.Metaphysician Undercover

    Through a bijection we can determine cardinal equivalence. If two sets X and Y have a bijection between them -- something that can be objectively determined -- we say they are cardinally equivalent. We still don't know what a cardinal number is. We only know that X and Y are cardinally equivalent.

    But are you saying that you do this without using cardinal numbers? What is cardinality without any cardinal numbers?Metaphysician Undercover

    There's cardinal equivalence without cardinal numbers. If there's a bijection between X and Y, then X and Y are cardinally equivalent. But we still haven't said what a cardinal is.

    It's a bit like saying that the score in a baseball game is tied -- without saying what the score is. Maybe that helps. Or in the classic example of bijective equivalence, I can put a glove on my hand and determine that the number of fingers on the glove is the same as the number of fingers on my hand, simply by matching up the glove-fingers to the hand-fingers bijectively. But that doesn't tell me whether the number of fingers is 4, 5, or 12. Only that the number of fingers is the same on the glove and on my hand, by virtue of matching the fingers up bijectively.

    So via establishing a bijection between the glove-fingers and the hand-fingers, I can say that the number of fingers is cardinally equivalent between the glove and my hand. But I still can't assign a particular cardinal number to it.


    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

    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.

    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

    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.

    And to create a cardinal number requires a bijection with ordinals, so ordinals are logically prior to cardinal numbers.Metaphysician Undercover

    So we have this notion of cardinal equivalence. We want to define a cardinal number. In the old days we said that a cardinal number was the entire class of all sets cardinally equivalent to a given one. In the modern formulation, we say that the cardinal number of a set is the least ordinal cardinally equivalent to some given set.

    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.

    .

    Where I have a problem is with the cardinality which is logically prior to the ordinal numbers.Metaphysician Undercover

    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.

    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 we should banish the word cardinality, because it's vague as to whether you mean cardinal equivalence or cardinal number. What kind of existence does a bijection between two sets have? Well a bijection is a particular kind of function, and functions have mathematical existence. In fact if X and Y are sets, and f is a function between them, then f is a set too. So whatever kind of mathematical existence sets have, that's the kind a bijection has.

    Some philosophers would say that functions have a higher "type" than sets, but we're not doing type theory, and in set theory everything is a set, at the same level. But if your mathematical ontology puts functions into a different level of existence than sets, then whatever level functions live in, that's what a bijection is. A bijection is just a kind of function.
  • TonesInDeepFreeze
    1.8k


    Well, at least thank you for not saying 'thank you'.
  • fishfry
    2.6k
    Well, at least thank you for not saying 'thank you'.TonesInDeepFreeze

    I said LOL because I was amused/charmed by your lengthy pre-apologies and disclaimers before providing your commentary on my technical points. I thought you went overboard but that you probably felt that you needed to go to those lengths to placate me. Which at that moment I found amusing. If that makes sense. But that's what was behind the LOL.
  • TonesInDeepFreeze
    1.8k
    I said LOL because I was amused/charmedfishfry

    I didn't get why you chose that clause in particular. I see now - it was just the nominated example.

    (Disclaimer yes, apology and placation no.)
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