An uncountable countable set
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From: Tony Orlow on 25 Aug 2006 22:02 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> Given a set x, can we always determine card(x)? >>> I'm guessing, but I'm pretty sure that there is not an algorithm that >>> will tell you the ordinal index of the aleph that is the cardinality of >>> any given set. >> Good guess, except I have no idea what your driving at. > > Of course you don't, because you are ignorant of even of the basics of > your own question. With the axiom of choice (which is one method), > every set has a cardinality. That cardinality is a cardinal number. And > that cardinal number is an aleph indexed by an ordinal. So I answered > your quesion: No, I don't think there is a procedure to determine which > cardinal is the cardinality of a given set. But in my first answer I > gave an even sharper formulation by pining the question down to the > ordinal index. Of course, for some sets, we can prove that a certain > cardinal is the cardinality that set; but I don't think there can be a > general procedure to produce an answer for any set that might be > submitted to test. > >>> First you need to define the set in set theory and prove that it exists >>> in set theory. Then, set theory would not be inconsistent for there not >>> being an algorithm to determine the ordinal index of the aleph that is >>> the cardinality of any given set. >> So, you're saying that identifying a set with no cardinality doesn't >> make set theory inconsistent? I think I saw a different opinion yesterday. > > You INSIST on twisting just about every answer given you. > > If it is a theorem that every set has a cardinality, then it would be > inconsistent if it were also a theorem that there exists a set without > a cardinality. But the fact that we don't have a procedure to always > announce what SPECIFIC cardinality a set has does not contradict any > theorem of set theory. However, if we have a set which we can prove does NOT have any cardinality within the system, then there's a hole, yes? > >>> Again, you're just yapping without regard for the specific definitions >>> of such things as 'consistent'. >> Consistent: Adj. Without contradiction. > > Yes, and your remarks were without sense of that definition, as > SPECIFICALLY you've been told about a million times that a theory is > inconsistent iff there is a contradiction in the theory, which means > that there is sentence and its negation that are both members of the > theory. So, is "every set has a cardinality, either finite or an indexed aleph" a theorem or not? You suggest that the axiom of choice makes it so. I say that the set of bit positions required by the binary naturals does not have such a cardinality. > > MoeBlee >
From: MoeBlee on 25 Aug 2006 22:03 Tony Orlow wrote: > So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I > never thought so. Those are notations that can be understood precisely only in context of the particular treatment in which they occur. I said that IN A SYSTEM that constructs the reals as either Dedekind cuts or as equivalence classes of Cauchy sequences, then the naturals and rationals are isomorphically embedded but are not actual subsets of the reals. (That is BASIC undergraduate mathematics. I mean, even I know that, and I don't even have an education in mathematics! It's appalling how ignorant you are WHILE you so categorically opine.) If the system and construction are of a different sort, then we'd have to evaluate upon the specifics of that system and construction. But, again, as I said, for contexts that do not need to be so pedantic, such notation is understood well enough that the equations you mentioned do of course hold. By the way, a while ago I posted a very rigorous explication of decimal notation in a post to you. > > Because it's more a philosophical issue or an issue of terminology > > outside the system. The purpose of set theory is not to address the > > question of what is and is not a number. Rather, among the purposes of > > set theory is to axiomatize and construct the various number systems > > that are of interest. > > So, how do you know when you're doing math, and when you've strayed into > other territory? That's a question for philosophy. I have my own views on what mathematics is, but I don't hold that my own concept of mathematics is definitive. > >>>> you don't see scientists accepting transfinitology either. > >>> Of course, you have a survey of scientists to support your claim. > >> Uh, yes, right here. Why don't you survey thise that object, regarding > >> what they do? > > > > That's quite an unscientific survey method of yours. > > Yo, man, it's empirical evidence. Ask around. Life's an experiment. It's anecdotal evidence only. Check it out. A scientific experiment is not just any life experience. > > Numbers are among the primary concern of mathematics. Set theory > > axiomatizes and constructs number systems. > > Which are the other primary concerns, if any? (how did we define > "number" again?) I don't define 'number' in a formal theory. But in informal discussion about mathematics I don't demur from using the word 'number' in its ordinary dictionary senses. Other concerns of mathematics are sets, relations, spaces, geometries, topologies, algebras, for example, and more. > I'll take as an "I dunno". Then you'll, AS USUAL, take incorrectly. MoeBlee
From: MoeBlee on 25 Aug 2006 22:19 Tony Orlow wrote: > > If it is a theorem that every set has a cardinality, then it would be > > inconsistent if it were also a theorem that there exists a set without > > a cardinality. But the fact that we don't have a procedure to always > > announce what SPECIFIC cardinality a set has does not contradict any > > theorem of set theory. > However, if we have a set which we can prove does NOT have any > cardinality within the system, then there's a hole, yes? If, using only first order logic with identity, you can prove from ZFC that there exists a set that does not have a cardinality, then you will have proven that ZFC (and just ZF for that matter, since ZFC is relatively consistent with ZF) is inconsistent. > So, is "every set has a cardinality, either finite or an indexed aleph" > a theorem or not? It is a theorem of ZFC. And every aleph is indexed, so you don't need to mention that for this purpose. And by the way, that reminds me that I made a mistake in not qualifying that only INFINITE sets have an aleph as their cardinality. Finite sets have a natural number as their cardinality and infinite sets have an aleph as their cardinality. And if I mentioned ZC before, it should have been ZFC, since (as far as I know, we also need the axiom schema of replacement to have all the ordinals we need). So, yes, it is a theorem of ZFC that every finite set has a natural number as its cardinality and every infinite set has an aleph as its cardinality. > You suggest that the axiom of choice makes it so. And I should have mentioned that the axiom schema of replacement also is need (as far as I know) in addition to the Z axioms that are not derivable from the axiom schema of replacement. > I > say that the set of bit positions required by the binary naturals does > not have such a cardinality. First, define, from the language of ZFC, every term you use in that formulation and every term you use to prove that formulation. Second, show an argument using only first order logic with identity and that proceeds only from axioms and already established theorems of ZFC. Then we'll talk about it. MoeBlee
From: MoeBlee on 25 Aug 2006 22:23 MoeBlee wrote: > With the axiom of choice (which is one method), > every set has a cardinality. That cardinality is a cardinal number. And > that cardinal number is an aleph indexed by an ordinal. CORRECTION: That should be, in ZFC, every set has a cardinality. That cardinality is either a natural number or an aleph (and, of course, every aleph is indexed by an ordinal).
From: MoeBlee on 25 Aug 2006 22:26
Tony Orlow wrote: > > Logic determines truth. Induction is more than just a form of proof. Logic determines VALIDITY. And induciton is indeed more than just a form of proof. But your concept of induction is uninformed, superficial, and confused. MoeBlee |