An uncountable countable set
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From: MoeBlee on 25 Sep 2006 11:24 Han de Bruijn wrote: > Absolute also means that e.g. an Alien Civilization can understand those > axiomatized theories. What makes you think that such will be the case? I can't predict what will be understood by aliens. But for certain systems, it is computable whether a string of symbols is or is not an axiom and it is computable whether a given string of symbols is or is not a proof. > They are all in Dutch. But maybe I'll take a look at it tonight. I'm glad to hear that there are dozens of textbooks on mathematical logic written in Dutch. The closest I have is van Dalen's in English and L.T.F. Gamut's in English. Maybe you might try one in English, such as Enderton's 'A Mathematical Introduction To Logic'. MoeBlee
From: Han.deBruijn on 25 Sep 2006 15:04 MoeBlee schreef: > Han de Bruijn wrote: > > Absolute also means that e.g. an Alien Civilization can understand those > > axiomatized theories. What makes you think that such will be the case? > > I can't predict what will be understood by aliens. But for certain > systems, it is computable whether a string of symbols is or is not an > axiom and it is computable whether a given string of symbols is or is > not a proof. I'm still flabbergasted why those difficult proofs as for Fermat's Last Theorem or the Poincare Conjecture are not proved then with the full power of modern computers. Han de Bruijn
From: Virgil on 25 Sep 2006 15:19 In article <1159185473.917531.157040(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Correct. 3 or III or {D,i,k} is a means which can be used to calculate > with. All those objects are numbers which can be used by an intelligent > being to count. Of course, also 3.000 is a number and "three", and > perhaps some intelligent being uses "kashdgaejh" to denote the number > three. There is some relativity. This ignores the axiom of extentionality. That one can use different sets of "pebbles" to tally, does not mean that the sets are the same. Several sets may all have the common property of being pairwise bijectable, but if any of their members are distinguishable from those of another set then the sets are equally distinguishable. > > > > You stated that you needed counting to determine the successor. That is > > false. The successor is defined without any reference to counting. > > The successor function *is* counting (+1). Not to those who can't count. Successorship does not require numbers, it only requires "next".
From: Virgil on 25 Sep 2006 15:30 In article <1159186907.615747.304410(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 1/3 is a number, properly defined, for instance, by the pair of numbers > 1,3 or 2,6 or 3,9 etc. But 0.333... is not properly defined because you > cannot index all positions, you cannot distinguish the positions of > this number from those with finite sequences (and you cannot > distinguish them from other infinte sequences which could exist, if one > could exist). Def: 0.333... = lim_{n -> oo} Sum_{k = 1..n} 1/3^n
From: Han.deBruijn on 25 Sep 2006 15:34
MoeBlee wrote: > Han de Bruijn wrote: > > They are all in Dutch. But maybe I'll take a look at it tonight. > > I'm glad to hear that there are dozens of textbooks on mathematical > logic written in Dutch. The closest I have is van Dalen's in English > and L.T.F. Gamut's in English. Hmm, I've been "somewhat" optimistic. Take that 12/3. One translated from French, one from German ... > Maybe you might try one in English, such as Enderton's 'A Mathematical > Introduction To Logic'. A meager result altogether. I have two books which are a translation from the English original. Maybe it's not the level you're expecting: Paul Halmos, "Naive Set Theory", Princeton 1960. E. Nagel, J.R. Newman, "Goedels Proof", New York, 1968. Han de Bruijn |