Cantor Confusion
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From: cbrown on 22 Oct 2006 15:30 mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > * we "can get to" (show the existence of) a set whose members satisfy > > the requirements described by omega in AoI, > > > > then we wouldn't "need" (have to separately assume) the *axiom* of > > infinity in order to talk about omega actually being a set. It would > > simply /logically follow/ as a *theorem* from the other axioms. > > Of course we cannot get to mega by counting. According to set theory we > get to omega by the limit: No, according to most set theories, we "get to" omega by simply assuming it exists. Set theory doesn't define limits at all. Topology and analysis define limits. > > lim [n-->oo] {1,2,3,...,n} = N. Here, one assumes you mean pointwise convergerence; and of course you are explicitly /assuming/ that N /is/ a set - otherwise, N has no meaning. > lim [n-->oo] {1 + 1/2^2 + 1/3^2 + ... + 1/n^2} = (pi^2)/6. Here, one assumes you mean using the usual metric topology; and that the toplogical space is the completed space of sequences which converge under that topology. > lim [t-->oo] X(t) = X(omega). > In what topology? Without specification, your statement is useless. lim n->oo f(n) is /not/ in general defined as f(omega). If it were, then your first limit would not be N; it would be N union {N}. > This limit is some existing entity which can be used, not something > that only can be approached as closely as you want. > That is the distinction between asserting "if a limit converges, then we accept that there always exist an element in our space to which it converges" (it can be used); as opposed to claiming "the limit converges, but possibly not to an element of the space". For example, if we restrict ourself to the rationals only, then the sequence you mention above regarding pi^2/6 "converges" in some sense, but not to a rational number - the rational numbers are not complete with respect to the usual metric. > > > > ZFC excluding AoI does /not/ say "you can get to omega" in the sense > > you are using "can get to" here. > > Therefore ZFC excluding INF is not sufficient to prove the existence of > numerical representations of irrational numbers, which in fact do not > exist. Don't misunderstand me: The ratio of circumference to diameter > of a circle is pi. But pi is not representable as a number. > So, pi is a number; but it is not representable as a number? > > Instead, ZFC /with/ AoI says, "since, > > if we are honest, we have to admit that you /can't/ 'get to' omega > > using the other axioms, we must therefore /assume/ omega's existence, > > in order to talk logically about arguments that assume omega is a set > > in the first place". > > In order to assume that an irrational number does exist with its omega > digits in decimal representation. First of all, there are other ways to describe irrationals such as sqrt(2) which don't rely on describing it as the limit of a sequence of approximations. For example, we can describe it as "that number which, when squared, equals 2" or "the number which is the ratio of the diagonal of a unit square to the number 1". Secondly, we don't require that the set of all natural numbers exist in order to define the limit of a sequence of rational numbers. We merely require a definition of what it means to be a natural number; so that we can say "n is a natural". Once we have done that, we can say things like "there is a natural m such that, for all n, if n is a natural with n > m, then f(n) < f(m)"; without opining either way on the validity of the statement "there exists a set containing all natural numbers". > > > > But so what? In a mathematical discussion amongst set theorists, you > > /don't have to/ accept that it's true or obvious or reasonable or > > sensible or accords with current dogma or is religious law or is > > required under punishment of banishment or /whatever/, that "omega is a > > set". > > > > Of course that assumes that you have /some/ axioms in mind: otherwise > > it is simply not a mathematical question whether your statements > > actually follow, one from the other, in your argument. It is instead an > > argument of philosophy. > > True mathematical questions can be answered by physical experiments. I find this philosophical stance quite puzzling, and extremely unsatisfying in its circularity; since generally physical experiments rely on deductions made using some pre-existing mathematical model. > For matheology you need belief or axioms. In order to get out of bed in the morning you need beliefs and assumptions - you need to believe, for example, that the floor will support your weight, and that you will not instead plunge into a fiery hell. In order to have a reasonable mathematical discussion, we need to /agree/ on what we are talking about. Those agreements are codified by axioms; and we agree that the logical conclusions derived from those axioms will be the area of our discourse. Anything else is philosophy; which is valuable, but not in and of itself mathematical. > > > > To clarify: suppose we agree that "since there is a set having the > > properties of omega, therefore ..." is not a valid for a correct > > argument. Do you have complaints about the remaining assumptions and > > axioms of ZFC? Or do you find them (and the logical conclusions that > > follow from them) to be acceptable as being true, obvious, logical, > > correct, etc. in mathematical discourse? > > Finite set theory is fine and true and useful . I wouldn't neccessarily call ZFC - AoI "finite set theory". We cannot deduce "there does not exist a Dedekind-infinite set" from ZFC - AoI; that too would need to be asserted as an axiom. > The axiom of choice > then is true too although it is no longer needed. You are assuming that ZFC - AoI implies all sets are finite; that is not the case. >The only fault, and > one of the most damaging in the scientific history of mankind, is the > introduction of actual infinity, i.e., finished infinity. > That is an opinion we don't share; it's certainly not a mathematical conclusion we can draw from the assumptions of ZFC - AoI. Cheers - Chas
From: Virgil on 22 Oct 2006 16:21 In article <1161503782.317444.80680(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > True mathematical questions can be answered by physical experiments. > For matheology you need belief or axioms. There is no physical experiment that proves that two plus two equals four.
From: Virgil on 22 Oct 2006 16:36 In article <1161516290.280024.6610(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > The identity of 1 and 0.999... in analysis is proven. The difference > > > 10^(-n) disappears for n --> oo (and not earlier!). The identity is not > > > proven in case of the diagonal number because it cannot be proven, > > > because of the lacking factor 10^(-n). Even for the digit with index n > > > --> oo the difference is as important as for the fist digit. > > > > Except for such cases as 1.000... = 0.999..., and others involving an > > endless string of 0's or an endless string of 9's, two reals differ if > > they differ in ANY digit, regardless of place value. > > Why not in such cases as 1.000... = 0.999...? Because in any base, b, there are countably many rationals of form a/b^n which have two representations in that base, one "ending" in an infinite string of 0's and the other in an infinite string of (b-1) digits. If one avoids both digits, then the expansion is unique, and any representation which differs from it in any digit position represents a different real. > You, as always, assert something just as you like it, without proof. Some things should be simple enough for even "Mueckenh" to sort out without formal proofs. > This 10^(-n) is the key to the inconsistency. > > Represent a number by 0.a_1,a_2,a_3,...,a_n,... with digit a_k eps > {0,1,2,3,...,9}. > > In the limit {n-->oo} we have 0.a_1,a_2,a_3,...,a_n,... = > 0.a_1,a_2,a_3,...,1+a_n,... > Therefore Cantor's diagonal argumen fails. If the "diagonal" is constructed out of only say 4's and 7's then any difference from it in any digit positions represents a different real number. > > > So a number whose > > decimal representation does not contain any occurrence of the digits 0 > > or 9 will differ in value from any other number if it differs at digit > > from the other number. > > If such a number really was defined, then the digits of pi could form a > natural number. Only in a "Mueckenh" world. > Stay with your belief. Makes much more sense than "Mueckenh"'s.
From: Virgil on 22 Oct 2006 16:40 In article <1161516809.429040.211720(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > Han de Bruijn wrote: > > > Bob Kolker wrote: > > > > > > > The are mathematical systems we know for sure are consistent since they > > > > have finite models. For example Just Plain Old Group Theory. No > > > > contradictions can be inferred from the group postulates simpliciter. > > > > > > True. And that is because the axioms of group theory actually serve as > > > a _definition_ of what a group _is_. Right? > > > > Just as the axioms of set theory define what a set is > > They don't define this at all. > > >, the axioms of the > > natural numbers define what a natural number is, > > This knowledge has been present over centuries until someone set out to > construct axioms to describe what was completerly superfluous because > everybody knew. Just like up until Galileo everybody knew that heavy objects fell faster? It has often turned out to be very useful to confirm, or refute, "what everybody knows".
From: Virgil on 22 Oct 2006 16:52
In article <1161517636.934369.301190(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > lim{n --> oo} 1/n = 0 proclaims "omega reached". > > > > It proclaims nothing of the sort. You want to read more in notation than > > is present. > > You want to read less than present and necessary. > lim{k --> oo} a_k = pi means omega reached. Or would you state that pi > belongs to Q like every a_k for natural k? According to one standard definition, lim{k --> oo} a_k = pi means for every positive epsilon there is a natural n such that if k >= n, then |a_k - pi | < epsilon. That definition nowhere implies that "omega" is reached, or even exists. Does "Mueckenh" have an alternate definition in mind? > > > > It is not. When giving a function from R to R, f(0) might exist, but > > f(oo) *never* exists. So asking for f(0) in the first case is a > > legitimate question. > > Not more legitimate than asking for f(oo) because at noon also > infinitely many transactions must have happened. Therefore the > distinction you make is false. Not hardly. f(t) as t increases towards 0 through a continuous set of values is quite different from g(n) as n increases without limit through a discrete set of values. > > > After the transformation you ask for f(oo), which > > is *not* legitimate. So the transformation does not simplify notation, > > it simply transforms the problem to an illegitimate problem. > > I hope you see your error. What alleged error it that? > Without the actual existence of omega both > problems with omega transactions are undefined. No standard limit definitions that I am aware of require the explicit existence of omega. Does "Mueckenh" have any references to definitions which do? |