Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 27 Jul 2005 13:36 Randy Poe said: > > > Tony Orlow (aeo6) wrote: > > Well, if this is how mathematicians feel about their study, and claim it has no > > connection to reality, then Cantorian set theorists really have no reason to > > claim that anything they say is more correct than any competing claims. > > As usual, you are confused. Your arguments deal with conclusions > which you draw FROM THE SAME AXIOMS as the competing, mainstream > theorems you denigrate. > > Mathematicians have every right to say that theorems deduced > from a given set of axioms by rigorous logic are more > correct UNDER THOSE AXIOMS than a competing, contradicting > set of conclusions someone claims arise UNDER THE SAME > AXIOMS. > > Some outside "reality" is not the standard. The basis of > argument in your eternal threads is the validity of > your conclusions UNDER A SPECIFIC SET OF AXIOMS AND > THE RULES OF DEDUCTION. > > - Randy > > Obviously, i am rejecting some of your axioms and assumptions, so I am not working with the same set as you are. Mathematics as a whole should be viewed as a system of axioms that should be consistent overall. That is the real litmus test of a given subsystem: external consistency within the broader field of all math. -- Smiles, Tony
From: Robert Low on 27 Jul 2005 13:36 malbrain(a)yahoo.com wrote: > Robert Low [incorrectly] wrote: >>But nobody has disagreed with that. The point of contention is >>whether the union over all n of S^n contains a finite string; >>it doesn't, but you claim it does. > I think you meant whether the union contains an INFINITE string. Absolutely: and I cancelled that message and sent the corrected one within ten seconds of posting it. Curse the efficiency of the internet.
From: Tony Orlow on 27 Jul 2005 13:39 malbrain(a)yahoo.com said: > Randy Poe wrote: > > Tony Orlow (aeo6) wrote: > > > Well, if this is how mathematicians feel about their study, and claim it has no > > > connection to reality, then Cantorian set theorists really have no reason to > > > claim that anything they say is more correct than any competing claims. > > > > As usual, you are confused. Your arguments deal with conclusions > > which you draw FROM THE SAME AXIOMS as the competing, mainstream > > theorems you denigrate. > > Ok. Perhaps you might give us the pre-set-theory version of the axiom > of induction. I presume it didn't just spring-up from nothing. Zeno's paradox is certainly an early form of inductive reasoning. > > > Mathematicians have every right to say that theorems deduced > > from a given set of axioms by rigorous logic are more > > correct UNDER THOSE AXIOMS than a competing, contradicting > > set of conclusions someone claims arise UNDER THE SAME > > AXIOMS. > > >From Webster's 1913 on-line dictionary: > Cor*rect" (k?r-r?kt"), a. [L. correctus, p. p. of corrigere to make > straight, to correct; cor- + regere to lead straight: cf. F. correct. > See Regular, Right, and cf. Escort.] Set right, or made straight; > hence, conformable to truth, rectitude, or propriety, or to a just > standard; not faulty or imperfect; free from error; as, correct > behavior; correct views > > It would seem that correct stems from Euclid's axioms. LOL good one. > karl m > > -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 13:41 malbrain(a)yahoo.com said: > Tony Orlow (aeo6) wrote: > > malbrain(a)yahoo.com said: > > > 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 > > > > > > > > In any case, sure, the program will spit out finite numbers, snce it is a > > finite machine running in finite time. If the machine had infinite capacity and > > infinite funtime, it could conceivably produce infinite results. > > I think you mean that the machine has finite runtime. That's what the > OPERATING SYSTEM enforces. My java-virtual-machines strike the program > dead with a BOLT OF LIGHTNING when it runs amok. karl m > > OOOPS!!! Well, MY computer only has limited funtime, unfortunately. Heh! Ummm, doesn't all that lightning damage the motherboard? -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 13:45
malbrain(a)yahoo.com said: > Tony Orlow (aeo6) wrote: > > I am not an expert in rings, nor am I going to sketch the axioms that you know > > better than I, nor does it help the conversation when you snip the original > > statement was repsonding to, which had soemthing to do with numbers being their > > own multiples of more than 1, or something. It didn't make sense in the context > > of normal quantitative addition. It was a vague guess as to what the idea was. > > > > The sum of an infinite number of 1's is infinite. That's all. > > Here's what Pascal had to say about his triangle in the 1600s: > > "Even though this proposition may have an infinite number of cases, I > shall give a very short proof of it assuming two lemmas. The first, > which is self evident, is that the proposition is valid for the second > row. The second is that if the proposition is valid for any row then it > must necessarily be valid for the following row. From this it can be > seen that it is necessarily valid for all rows; for it is valid for the > second row by the first lemma; then by the second lemma it must be true > for the third row, and hence for the fourth, and so on to infinity" > > karl m > > That sounds pretty inductive to me. Pascal was alright. You know, Pascal's triangle is a table of the number of boundary features of each dimension on a triaguloid of a given dimension? Like, how many tetrahedrons are on the boundary of a 9D triaguloid......like that. Just an aside. -- Smiles, Tony |