Uncountability of the Rationals?
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From: Jesse F. Hughes on 20 Jun 2010 08:59 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 18, 8:52 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Tony Orlow <t...(a)lightlink.com> writes: >> > On Jun 16, 11:00 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Transfer Principle <lwal...(a)lausd.net> writes: >> >> > How about this: >> >> >> > An algebraic function is a real-valued function which is the >> >> > composition of finitely many real-valued polynomial, radical, >> >> > rational, exponential, and logarithmic functions, and whose >> >> > inverse (or at least the real-valued branches thereof) is >> >> > also the composition of finitely many polynomial, radical, >> >> > rational, exponential, and logarithmic functions. >> >> >> Yes, that's explicit. >> >> >> And, who knows, it might be what Tony meant. Perhaps he'll say so. >> >> > That list probably covers the gamut, at least for now. So, I guess I >> > didn't misuse "algebraic" after all. >> >> Yes, you did misuse "algebraic". In my experience, an algebraic >> function is one which preserves certain algebraic structure. >> >> But no matter. We'll assume that Walker's definition of "algebraic >> bijection" is what you "probably" (probably?) meant. >> >> I guess it will follow that the set P of primes has no size, since there >> is no algebraic bijections between P and N+? > > http://primes.utm.edu/howmany.shtml > "The Prime Number Theorem: The number of primes not exceeding x is > asymptotic to x/log x." > So, let's say tav/log(tav). No, let's not just make things up. You have said, up 'til now, that the size of a subset S of N+ (I'm still not sure what the "+" signifies) depends on having an "algebraic bijection" between S and N+. I don't see any such function between P and N+. Consistency is no vice, Tony. -- Jesse F. Hughes Baba: Spell checkers are bad. Quincy (age 7): C-H-E-K-E-R-S A-R-E B-A-D.
From: Tony Orlow on 20 Jun 2010 09:52 On Jun 20, 8:59 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > On Jun 18, 8:52 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Tony Orlow <t...(a)lightlink.com> writes: > >> > On Jun 16, 11:00 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> >> Transfer Principle <lwal...(a)lausd.net> writes: > >> >> > How about this: > > >> >> > An algebraic function is a real-valued function which is the > >> >> > composition of finitely many real-valued polynomial, radical, > >> >> > rational, exponential, and logarithmic functions, and whose > >> >> > inverse (or at least the real-valued branches thereof) is > >> >> > also the composition of finitely many polynomial, radical, > >> >> > rational, exponential, and logarithmic functions. > > >> >> Yes, that's explicit. > > >> >> And, who knows, it might be what Tony meant. Perhaps he'll say so. > > >> > That list probably covers the gamut, at least for now. So, I guess I > >> > didn't misuse "algebraic" after all. > > >> Yes, you did misuse "algebraic". In my experience, an algebraic > >> function is one which preserves certain algebraic structure. > > >> But no matter. We'll assume that Walker's definition of "algebraic > >> bijection" is what you "probably" (probably?) meant. > > >> I guess it will follow that the set P of primes has no size, since there > >> is no algebraic bijections between P and N+? > > >http://primes.utm.edu/howmany.shtml > > "The Prime Number Theorem: The number of primes not exceeding x is > > asymptotic to x/log x." > > So, let's say tav/log(tav). > > No, let's not just make things up. > > You have said, up 'til now, that the size of a subset S of N+ (I'm still > not sure what the "+" signifies) depends on having an "algebraic > bijection" between S and N+. I don't see any such function between P > and N+. > > Consistency is no vice, Tony. > > -- > Jesse F. Hughes > > Baba: Spell checkers are bad. > Quincy (age 7): C-H-E-K-E-R-S A-R-E B-A-D.- Hide quoted text - > > - Show quoted text - I didn't say this situation was handled by IFR and ICI. I even stated that some sets might not have set sizes more specific than cardinality in some cases. However, given the aymptotic relationship discovered by others, I would guess that there is justification for quantifying this set with this formula. Is there a relatively simple inverse to x/log x? If so, it might be used to estimate the locations of primes, but probably not. TOny
From: Jesse F. Hughes on 20 Jun 2010 10:26 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 20, 8:59 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Tony Orlow <t...(a)lightlink.com> writes: >> > On Jun 18, 8:52 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Tony Orlow <t...(a)lightlink.com> writes: >> >> > On Jun 16, 11:00 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> >> Transfer Principle <lwal...(a)lausd.net> writes: >> >> >> > How about this: >> >> >> >> > An algebraic function is a real-valued function which is the >> >> >> > composition of finitely many real-valued polynomial, radical, >> >> >> > rational, exponential, and logarithmic functions, and whose >> >> >> > inverse (or at least the real-valued branches thereof) is >> >> >> > also the composition of finitely many polynomial, radical, >> >> >> > rational, exponential, and logarithmic functions. >> >> >> >> Yes, that's explicit. >> >> >> >> And, who knows, it might be what Tony meant. Perhaps he'll say so. >> >> >> > That list probably covers the gamut, at least for now. So, I guess I >> >> > didn't misuse "algebraic" after all. >> >> >> Yes, you did misuse "algebraic". In my experience, an algebraic >> >> function is one which preserves certain algebraic structure. >> >> >> But no matter. We'll assume that Walker's definition of "algebraic >> >> bijection" is what you "probably" (probably?) meant. >> >> >> I guess it will follow that the set P of primes has no size, since there >> >> is no algebraic bijections between P and N+? >> >> >http://primes.utm.edu/howmany.shtml >> > "The Prime Number Theorem: The number of primes not exceeding x is >> > asymptotic to x/log x." >> > So, let's say tav/log(tav). >> >> No, let's not just make things up. >> >> You have said, up 'til now, that the size of a subset S of N+ (I'm still >> not sure what the "+" signifies) depends on having an "algebraic >> bijection" between S and N+. I don't see any such function between P >> and N+. >> >> Consistency is no vice, Tony. >> >> -- >> Jesse F. Hughes >> >> Baba: Spell checkers are bad. >> Quincy (age 7): C-H-E-K-E-R-S A-R-E B-A-D.- Hide quoted text - >> >> - Show quoted text - > > I didn't say this situation was handled by IFR and ICI. I even stated > that some sets might not have set sizes more specific than cardinality > in some cases. However, given the aymptotic relationship discovered by > others, I would guess that there is justification for quantifying this > set with this formula. Is there a relatively simple inverse to x/log > x? If so, it might be used to estimate the locations of primes, but > probably not. So, as I first suggested, according to your "theory" of sizes, P has no size. You disagreed, saying we should just "say" that the size is tav/log(tav). Do you now agree that P has no size, according to your own notion of size? -- Jesse F. Hughes "Contrariwise," continued Tweedledee, "if it was so, it might be, and if it were so, it would be; but as it isn't, it ain't. That's logic!" -- Lewis Carroll
From: Tony Orlow on 20 Jun 2010 10:41 On Jun 20, 10:26 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > On Jun 20, 8:59 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Tony Orlow <t...(a)lightlink.com> writes: > >> > On Jun 18, 8:52 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> >> Tony Orlow <t...(a)lightlink.com> writes: > >> >> > On Jun 16, 11:00 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> >> >> Transfer Principle <lwal...(a)lausd.net> writes: > >> >> >> > How about this: > > >> >> >> > An algebraic function is a real-valued function which is the > >> >> >> > composition of finitely many real-valued polynomial, radical, > >> >> >> > rational, exponential, and logarithmic functions, and whose > >> >> >> > inverse (or at least the real-valued branches thereof) is > >> >> >> > also the composition of finitely many polynomial, radical, > >> >> >> > rational, exponential, and logarithmic functions. > > >> >> >> Yes, that's explicit. > > >> >> >> And, who knows, it might be what Tony meant. Perhaps he'll say so. > > >> >> > That list probably covers the gamut, at least for now. So, I guess I > >> >> > didn't misuse "algebraic" after all. > > >> >> Yes, you did misuse "algebraic". In my experience, an algebraic > >> >> function is one which preserves certain algebraic structure. > > >> >> But no matter. We'll assume that Walker's definition of "algebraic > >> >> bijection" is what you "probably" (probably?) meant. > > >> >> I guess it will follow that the set P of primes has no size, since there > >> >> is no algebraic bijections between P and N+? > > >> >http://primes.utm.edu/howmany.shtml > >> > "The Prime Number Theorem: The number of primes not exceeding x is > >> > asymptotic to x/log x." > >> > So, let's say tav/log(tav). > > >> No, let's not just make things up. > > >> You have said, up 'til now, that the size of a subset S of N+ (I'm still > >> not sure what the "+" signifies) depends on having an "algebraic > >> bijection" between S and N+. I don't see any such function between P > >> and N+. > > >> Consistency is no vice, Tony. > > >> -- > >> Jesse F. Hughes > > >> Baba: Spell checkers are bad. > >> Quincy (age 7): C-H-E-K-E-R-S A-R-E B-A-D.- Hide quoted text - > > >> - Show quoted text - > > > I didn't say this situation was handled by IFR and ICI. I even stated > > that some sets might not have set sizes more specific than cardinality > > in some cases. However, given the aymptotic relationship discovered by > > others, I would guess that there is justification for quantifying this > > set with this formula. Is there a relatively simple inverse to x/log > > x? If so, it might be used to estimate the locations of primes, but > > probably not. > > So, as I first suggested, according to your "theory" of sizes, P has no > size. You disagreed, saying we should just "say" that the size is > tav/log(tav). > > Do you now agree that P has no size, according to your own notion of > size? Yes, the size of P is not derivable given the methods I have suggested. However, I would maintain that if lim(x->oo, y->x: P(y)/N+ (y)) = y/log y, then in this sense ICI applies, even if IFR is irrelevant. Do you see the distinction? Happy Father's Day! Tony > > -- > Jesse F. Hughes > "Contrariwise," continued Tweedledee, "if it was so, it might be, and > if it were so, it would be; but as it isn't, it ain't. That's logic!" > -- Lewis Carroll- Hide quoted text - > "The whole of science is nothing more than a refinement of everyday thinking." - Einstein Tony
From: Jesse F. Hughes on 20 Jun 2010 11:53
Tony Orlow <tony(a)lightlink.com> writes: > Yes, the size of P is not derivable given the methods I have > suggested. However, I would maintain that if lim(x->oo, y->x: P(y)/N+ > (y)) = y/log y, then in this sense ICI applies, even if IFR is > irrelevant. Do you see the distinction? Well, I first notice that P(y) and N+(y) are so far utterly undefined and thus you haven't expressed yourself clearly. As far as ICI, I'm afraid I don't recall what it is, but I'm pretty sure that you haven't said anything that implies the conclusion you've just drawn. -- "Sure, maybe I have a tiresome task that is nearly impossible, but part of who I am is an endless amount of energy as long as there is hope. Without hope, I find that I start to lose focus, and feel, just, well, hopeless." -- James S. Harris |