Cantor Confusion
in [Math]
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From: Virgil on 9 May 2007 17:42 In article <1178731819.918168.255140(a)h2g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Mai, 20:00, Virgil <vir...(a)comcast.net> wrote: > > > And perhaps, as a non-mathematician, one might even say an > > anti-mathematician, WM is incompetent to judge the value of mathematical > > definitions to mathematics. > > If you call yourself a mathematician, then I will agree to the honour > of being an anti-mathematician. > > > > I attended Harvard and Oxford, among others. > > Not studying mathematics there, I hope. Studying math at both. > At least not learning it. Learning a good deal more of it than WM has learnt, at all events. I could hardly have learned less. Or > did they teach you infinite definitions there? Probably they did not > get ready yet? > > > > > You need it taught by anyone at all with any more mathematical > > competence than you have yourself. Only the ignorant laugh at their own > > ignorance. > > Then try to stop it. Do you know the definition of a binary relation > by H&J? > > 2.1 Definition A set R is a binary relation if all elements of R > are ordered pairs, i.e., if for any z � R there exist x and y such > that z = (x, y). > > 2.3 Definition Let R be a binary relation. > (a) The set of all x which are in relation R with some y is called the > domain of R and denoted by dom R = {x | there exists y such that xRy}. > dom R is the set of all first coordinates of ordered pairs in R. > (b) The set of all y such that, for some x, x is in relation R with y > is called the range of R, denoted by ran R. So ran R = {y | there > exists x such that xRy}. > > Can you find the words domain and range in this text? > I find 'domain' and 'range' to be defined in terms of a set of ordered pairs, not the reverse, as WM has been trying to imply.
From: Ralf Bader on 9 May 2007 18:12 WM wrote: > On 9 Mai, 20:25, Virgil <vir...(a)comcast.net> wrote: >> In article <1178716861.173911.234...(a)y5g2000hsa.googlegroups.com>, > >> WM has no power to constrain mathematics to what he thinks it should be. >> >> That it is not what WM thinks it should be is by now apparent, but that >> it is quite productive anyway is equally apparent > > Producing what? Yes, the working mathematicians, they are quite > productive, but they do not need transfinite set theory. Transfinite > set theory is the most useless science ever created. This may be. However, there are non-sciences which are far more useless than set-theory. For example an idiocy called "Matherealism" by its inventor. Can you name any mathematician or philosopher of stature who even took notice of "Matherealism"? (Publicly, of course, no ramblings about asserted private messages of unknown content)
From: Virgil on 9 May 2007 18:14 In article <1178736555.193942.235110(a)u30g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 9 Mai, 17:39, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > Why then did you write the unnecessary paths? > > > > > Why then did you write the unnecessary paths? > > Why then did you write the unnecessary paths? To try and pierce that veil of ignorance and self-deceit that WM is hiding behind!
From: Virgil on 9 May 2007 18:16 In article <1178737042.043976.162390(a)e65g2000hsc.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 9 Mai, 18:04, William Hughes <wpihug...(a)hotmail.com> wrote: > > On May 9, 11:27 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 9 Mai, 16:46, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > On May 9, 10:34 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > I gave a bijection between the set of nodes and the set of branching- > > > > > offs of paths bunches. As there can be not more results of branching- > > > > > offs than branching-offs, the task is done. > > > > > > There are only a countable number of results of branching-off. > > > > > > Whoops, an inifinite path is not a result of branching-off > > > > > No? Is there no infinite branching off in 0,010101...? > > > > No. There are an unbounded number of branchings, each of which takes > > place at a finite position. There is no branching which takes place at > > an infinite position. > > Correct. The infinite path representing 1/3 is the union of all finite > paths of this kind: 0.0, 0.01, 0.010, ... > It has not a single branching more than the union of these paths. It > has not a single node more than the union of these paths. Therefore > this path is the result of aleph_0 branching-offs. But each branching off separates it from uncountably many other paths that have not been previously separated from it.
From: MoeBlee on 9 May 2007 19:00
On May 9, 12:17 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > How do you define a set, unless you cannot specify the elements > belonging to it? (I suppose 'cannot' is a typo and that you mean 'can'.) You're asking that question yet you claim to understand set theory (!). There are at least two senses of 'definition' to consider (not necessarily in order of importance): 1. For an element of the universe of a model for a language, we can define that element in the language if there is a term in the language that maps, per the model, per the recursively defined function from the set of terms into the model, to that element. Or we can generalize to allow that an element of the universe of a model for the language is defined by a formula P such that per the model, per the recursively defined function from the set of terms into the model, that element element alone satisfies P. That's quite awkard to say all in one English sentence, but were I to give it in the actual mathematics of model theory, it would be perspicacious. 2. If, in a theory, we have a theorem E!xP, where P is a formula in the language of the theory (call this 'the source language'), then we can extend the source language through a definitional axiom of the form c = x <-> P (where 'c' is a constant symbol that is not a symbol of the source language). > > Define "formula". Until it is defined, it has no mathematical > > significance, and it hasn't yet been defined. > > A formula is a rule or prescription specifying or determining which > element of the domain is mapped on which element of the range. The > formula in its widest sense can even consist of several expressions. > In case every element of the domain has to be mapped by its own > expression, the formula can even be a catalogue or set. Again, that may be some informal notion you've gleaned from some book or another, but it is not at all the notion nor definition of 'formula' that is used in mathematical logic or in ordinary textbooks in set theory. You are very much in need of learning the inductive definitions of such terms as 'formula' and 'formula of set theory'. A good place to start, I think, is with Kalish, Montague, and Mar's 'Logic: Techniques Of Formal Reasoning', which will give you the basic skills in the predicate calculus that you need; then you'd be in a better position to study your Hrbacek & Jech; and around that time you could study Enderton's 'A Mathematical Introduction To Logic' for a good solid understanding of mathematical logic such as the notions of 'definition' that I merely synopsized in this post. Also, Suppes's 'Introduction To Logic' has the best introductory discussion of the subject of mathematical definitions that I have seen. And an abbreviated version of that discussion is in Suppes's 'Axiomatic Set Theory', which is among the best of the set theory textbooks and available for a great price in paperback. > > > > And the formal definition requires a function to be a set of ordered > > > > pairs (a relation), so that it is first of all a SET, which is quite > > > > different from being merely a rule.- > > > > Of course it is not merely a rule. It is a rule and two sets, which > > > are connected by this rule. > > > Not in mathematics. In mathematics, it is a relation with certain > > special properties. > > Do you know what a relation is? Do you know that there is also the > domain and the range required? Every function, indeed every relation (note that where Hrbacek & Jech say 'binary relation', most other authors, such as those I mentioned previously, say 'relation'), indeed every set (though Hrbacek & Jech, unlike certain other authors, give conditional definitions for 'domain' and 'range' applied only to relations) has a unique domain and a unique range; but the mere DEFINITION of 'function' does not require mention of whatever the domain and range of any particular function might be. Let's go over it one more time, since you are such an OBSTINATELY slow learner: Axiom: y=z <-> Ax(xez <-> xey) Definition: {r s} = x <-> Ay(yex <-> (y=r v y=s)) Definition: {r} = {r r} Definition: <r s> = {{r} {r s}} Definition: p is an ordered pair <-> Ers p=<r s> Definition: f is a relation <-> Apef p is an ordered pair Definition: f is a function <-> (f is a relation & Axyz((<x y>ef & <x z>ef ) -> y=z)) So, even though we usually will have defined 'domain' and 'range' before we define function, it is not necessary that we do. The definition of 'function' stands without having to have defined 'domain' and 'range'. Moreover, ALL of our definitions revert to the primitive language so that, for example, we don't even need to define 'relation' to define 'function' since where 'is a relation' occurs in the definition of 'function' we could substitute the definition of 'relation', and then substitute 'ordered pair' etc. Got it now? MoeBlee |