Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 26 Jul 2005 12:26 Dik T. Winter wrote: > In article <1122347583.518181.245300(a)g14g2000cwa.googlegroups.com> malbrain(a)yahoo.com writes: > > > > The C language is defined by the C standard, as defined by ISO. There > > are no "unbounded" standard types in the C language. karl m > > Who is talking about C? Of the billions of computer systems deployed since the micro-computer revolution, the overwhelming majority are programmed with C. karl m
From: briggs on 26 Jul 2005 12:31 In article <1122393788.077928.129590(a)o13g2000cwo.googlegroups.com>, malbrain(a)yahoo.com writes: > Daryl McCullough wrote: > >> Nothing in mathematics is excepted without question. Not by >> mathematicians, anyway. > > I think you meant ACCEPTED. See Barb's post for a discussion of > EXCEPTED. > >> Yes, it is certainly the case that *if* you can prove by >> induction "forall x, Phi(x)", *then* you can write a >> corresponding recursive function that given a number n, >> produces a proof of Phi(n). Nobody disputes that. What >> people are disputing is your bizarre belief that proving >> "forall x, Phi(x)" by induction means that you have proved >> Phi(0), Phi(1), Phi(2), ... It means that you *can* prove >> all those infinitely many statements, not that you *have*. > > Sorry, but the axiom states that you HAVE INDEED proved your assertion > for each and every n when its conditions are satisfied. The axiom makes no statement about what you *can* prove. The axiom makes no statement about what you HAVE INDEED proved. The axiom states that the _PROPERTY HOLDS_ for each and every n given that the property holds for 0 and that whenever the property holds for i, the property holds for S(i). Daryl appears to be attributing to Tony Orlow the belief that an inductive proof is equivalent to going through an infinite loop, manually proving each of Phi(1), Phi(2), ... and finally, magically, getting done and thereby concluding that Phi(n) holds for all n. [And also, since he finished the infinite loop, Tony can squint and decide that Phi(oo) also holds] To the extent that what Tony Orlow thinks can be rationally characterized, I think Daryl is on the money with his diagnosis. John Briggs
From: David Kastrup on 26 Jul 2005 12:45 malbrain(a)yahoo.com writes: > Tony Orlow (aeo6) wrote: >> malbrain(a)yahoo.com said: >> > Virgil wrote: >> > > In article <MPG.1d4863d52071fde5989f51(a)newsstand.cit.cornell.edu>, >> > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> > >> > > > and that I was trying to prove >> > > > things about sets, not numbers, which is also bullshit, since I was >> > > > proving a property regarding a set DEFINED by a natural number, which >> > > > is ultimately a property of that number. >> > > >> > > But the set N is not defined by any one natural number >> > >> > Under Tony's theory, the number representing N is defined by a string >> > of an infinite number of ones. Yes, more than one Turing machine can >> > produce this. karl m >> > >> > >> Actually there are two ways to look at it. In unsigned binary, yes, >> an infinite number of 1's is the largest number possible. Since we >> start with all 0's representing 0, the size of the set, N, will be >> one more than 111...111. It will be 000...001:000...000, or one >> unit infinity. > > There's already a STANDARD method for coding real numbers using the > integers. It's published by the Institute of Electrical and > Electronic Engineering. Uh, no. That's a method for coding floating point numbers. That's quite something else. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: stephen on 26 Jul 2005 12:51 In sci.math malbrain(a)yahoo.com wrote: > stephen(a)nomail.com wrote: >> In sci.math malbrain(a)yahoo.com wrote: >> > stephen(a)nomail.com wrote: >> >> In sci.math malbrain(a)yahoo.com wrote: >> >> > Barb Knox wrote: >> >> >> In article <1122338688.718048.162860(a)g47g2000cwa.googlegroups.com>, >> >> >> malbrain(a)yahoo.com wrote: >> >> >> >> >> >> >Barb Knox wrote: >> >> >> >> In article <MPG.1d4ecd45545679a8989f6b(a)newsstand.cit.cornell.edu>, >> >> >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> >> >> >> [snip] >> >> >> >> >> >> >> >> >keep in mind that >> >> >> >> >inductive proof IS an infinite loop, so that incrementing in the loop >> >> >> >> >creates >> >> >> >> >infinite values, and the quality of finiteness is not maintained over those >> >> >> >> >infinite iterations of the loop. >> >> >> >> >> >> >> >> Using your computational view, consider the following infinite loop >> >> >> >> (using some unbounded-precision arithmetic system similar to >> >> >> >> java.math.BigInteger): >> >> >> >> >> >> >> >> for (i = 0; ; i++) { >> >> >> >> println(i); >> >> >> >> } >> >> >> >> >> >> >> >> Now, although this is an INFINITE loop, every value printed will be >> >> >> >> FINITE. Right? >> >> >> > >> >> >> >Not so fast. The behaviour of incrementing i after it reaches INT_MAX >> >> >> >is undefined. >> >> >> >> >> >> Not so fast yourself. You missed the part about "unbounded-precision >> >> >> arithmetic system", which has no max. >> >> >> >> > Sorry, but the C standard admits no such system. INT_MAX must be >> >> > declared by the implementation. There is no room for exceptions. karl >> >> > m >> >> >> >> Why are you talking about C? >> >> > Because C is "better" defined than java, and the example is written in >> > C. karl m >> >> The example is not written in C. It is perfectly legal >> Java code, and C++ code, and C# code, and Javascript. >> Given that 'println' is not a standard C function, but >> it is a standard Java function, and the author identified >> the example as Java, it is pretty clear the example was written >> in Java. > I really don't want to get into a discussion of why the C standard > doesn't allow EXCEPTIONS when considering the infinite. Perhaps later > this afternoon the sky will open up a little. karl m Again, why are you talking about C? The example was written in Java, and clearly idenfified as Java. The C standard is irrelevant in a discussion about Java. Stephen
From: malbrain on 26 Jul 2005 13:05
stephen(a)nomail.com wrote: > In sci.math malbrain(a)yahoo.com wrote: > > stephen(a)nomail.com wrote: > >> In sci.math malbrain(a)yahoo.com wrote: > >> > stephen(a)nomail.com wrote: > >> >> In sci.math malbrain(a)yahoo.com wrote: > >> >> > Barb Knox wrote: > >> >> >> In article <1122338688.718048.162860(a)g47g2000cwa.googlegroups.com>, > >> >> >> malbrain(a)yahoo.com wrote: > >> >> >> > >> >> >> >Barb Knox wrote: > >> >> >> >> In article <MPG.1d4ecd45545679a8989f6b(a)newsstand.cit.cornell.edu>, > >> >> >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > >> >> >> >> [snip] > >> >> >> >> > >> >> >> >> >keep in mind that > >> >> >> >> >inductive proof IS an infinite loop, so that incrementing in the loop > >> >> >> >> >creates > >> >> >> >> >infinite values, and the quality of finiteness is not maintained over those > >> >> >> >> >infinite iterations of the loop. > >> >> >> >> > >> >> >> >> Using your computational view, consider the following infinite loop > >> >> >> >> (using some unbounded-precision arithmetic system similar to > >> >> >> >> java.math.BigInteger): > >> >> >> >> > >> >> >> >> for (i = 0; ; i++) { > >> >> >> >> println(i); > >> >> >> >> } > >> >> >> >> > >> >> >> >> Now, although this is an INFINITE loop, every value printed will be > >> >> >> >> FINITE. Right? > >> >> >> > > >> >> >> >Not so fast. The behaviour of incrementing i after it reaches INT_MAX > >> >> >> >is undefined. > >> >> >> > >> >> >> Not so fast yourself. You missed the part about "unbounded-precision > >> >> >> arithmetic system", which has no max. > >> >> > >> >> > Sorry, but the C standard admits no such system. INT_MAX must be > >> >> > declared by the implementation. There is no room for exceptions. karl > >> >> > m > >> >> > >> >> Why are you talking about C? > >> > >> > Because C is "better" defined than java, and the example is written in > >> > C. karl m > >> > >> The example is not written in C. It is perfectly legal > >> Java code, and C++ code, and C# code, and Javascript. > >> Given that 'println' is not a standard C function, but > >> it is a standard Java function, and the author identified > >> the example as Java, it is pretty clear the example was written > >> in Java. > > > I really don't want to get into a discussion of why the C standard > > doesn't allow EXCEPTIONS when considering the infinite. Perhaps later > > this afternoon the sky will open up a little. karl m > > Again, why are you talking about C? The example was written > in Java, and clearly idenfified as Java. The C standard > is irrelevant in a discussion about Java. No, it's not a valid java program. It doesn't cover itself for the BOLTS OF LIGHTNING that we can expect from the OPERATING SYSTEM it's deployed under. C doesn't use LIGHTNING -- it requires that INT_MAX be declared ahead of time by the implementation. karl m |