Galileo's Paradox
in [Math]
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From: Tony Orlow on 4 Dec 2006 12:33 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>> > > <snip> > >> Your other question about 3 needs to be thought about a bit. You could >> ask Dave Rusin, He confirmed a couple of years ago it was a simple >> repeating bit pattern. I suppose I need to do my own legowrk in that >> area now. >> > > Usually what we mean when we say "(some set of operations on a base > set) generates (some other set)" is "by a finite number of applications > of the rules on the base set". But no /finite/ number of applications > of your rules will generate "3". So you need to define a limit, in > addition to your two generating rules. Right. I need to define an algorithm such that one can, within some finite number of steps, get an H-riffic value arbitrarily close to any given real value. This is a bit difficult with these numbers, which are a superset of tetrations. It may take me a while, though I am sure there's someone out there with more experience than me in such things. Or, maybe not... > > (Unless you want to define your own personal meaning to "generates"; in > which case you will just be confusing people :) ). > > Note that your generator certainly will not produce -1, whether we > allow limits or not; so really the question is does it generate (in the > limit) all non-negative reals (R+)? Right. This form of the H-riffics produces the POSITIVE REALS. If you want negatives, we can change rule 2 to: E x -> E 2^x ^ E -2^x You can also define H-riffics within (0,1) if you want. > > To prove this, you need to prove that the closure under finite > applications of your rules is dense in R+; otherwise, there will be > gaps. Yes, I need to "prove" that. It's true, but not that simple to prove. I'll get back to that. > > Without limits, there are only a countable set of values generated; so > this would clearly fail to generate all reals. Well, it doesn't fail in that regard, though there will be H-riffics, like 3, which require infinite strings, like 1/3 in decimal. That's to be expected. > > Cheers - Chas > Tony
From: Eckard Blumschein on 4 Dec 2006 12:35 On 12/1/2006 9:50 PM, Virgil wrote: > In article <45700139.40908(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/30/2006 1:34 PM, Bob Kolker wrote: >> >> >> > Giants, such as Hilbert, welcomed Cantorian mathematics. >> >> It was David Hilbert who declared Cantor's CH the first out of 23 >> problems of mathematics, and he did so in 1900 at 2nd International >> Congress of Mathematicians when he was 38 years old, full professor for >> 7 years and chair of mathematics at U of Goettingen for 5 years. >> Cantor was founder and president of the Society. Both Hilbert and Cantor >> benefitted from their close friendship with Hurwitz. >> Because Hilbert successfully applied axioms to geometry in 1999, and set >> theory did not have a reliable basis, he may have felt in position and >> was perhaps invited to provide axioms for set theory, too. >> However, despite of beeing desined for this task, he did lower his >> standard and invent the masterpiece of delusion which has been ascribed >> to Zermelo: claiming the existence of infinite sets. > > The claim of existence of points and lines, etc. in Hilbert's > axiomatization of geometry relies on making equally unverifiable > assumptions of infiniteness. Infinity is an acient concept and parallels continuum. There is nothing to niggle about it.
From: Bob Kolker on 4 Dec 2006 12:41 Lester Zick wrote: > > Such as the integration of points into lines, Bob? 1. That is not at the heard of set theory. 2. It is not a contradiction. A possible model for Hilbert's axiomatization has lines as sets of points. Bob Kolker
From: Eckard Blumschein on 4 Dec 2006 12:44 On 12/1/2006 9:46 PM, Virgil wrote: > In article <456FEEEF.5070409(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/30/2006 1:32 PM, Bob Kolker wrote: >> > Eckard Blumschein wrote: >> > >> >> >> >> >> >> I consider Dedekind wrong, and he admitted to have no evidence in order >> >> to justify his basic idea. >> > >> > What sort of evidence? Surely not empirical evidence. Mathematics done >> > abstractly has no empirical content whatsoever. >> > >> > Bob Kolker >> >> Serious mathematicans have to know the pertaining confession. Dedekind >> wrote: "bin ausserstande irgendeinen Beweis f�r seine Richtigkeit >> beizubringen". In other words, he admitted being unable to furnish any >> mathematical proof which could substantiate his basic assumption. > > That conclusion assumes something not in evidence, that no one else has > been able to do what Dedekind said he had not done. Yes. It is about as impossible as to get younger. >> Consequently, any further conclusion does not have a sound basis. >> Dedekind's cuts are based on guesswork. > > So are everyone else's equally based on "guesswork", as without ASSUMING > something, one cannot deduce anything. The ideas by Archimedes and Aristotele, by Galilei and Spinoza, and by many others have proven very fruitful while Dedekind and Cantor just created something impressive. It demands to swallow countability of the uncountables, more than infinitely many numbers, transfinite integers, and the like, being altogether proven futile but an ongoing source of futile confusion about CH, AC, and axioms on demand.
From: Michael Stemper on 4 Dec 2006 13:12
In article <3stjm255vbgdfdrh9jdvrmbecu99perr0i(a)4ax.com>, Six Letters writes: >On 24 Nov 2006 15:08:20 GMT, richard(a)cogsci.ed.ac.uk (Richard Tobin) wrote: >>In article <f41em2h1s4dv1qm70ntnv2jped6qekfd7s(a)4ax.com>, <Six Letters> wrote: >>>> (a) sets in 1-1 correspondence are the same size and >>>> (b) proper subsets are smaller than their supersets >>> >>>Exactly. It's throwing out (a) that I am trying to explore. >> >>This will result in bizarre consequences. For example, there will be >>more decimal strings representing integers than binary strings, even >>though they represent the same integers. > > On second thoughts I am not sure that I understand that. I rather >suspect there are bizarre consequences, but I'd be really happy if you'd >elaborate a little bit. It's pretty simple. The first eleven positive integers are represented in decimal as: 1 2 3 4 5 6 7 8 9 10 11. The first three positive integers are represented in binary as: 1 10 11. How should a map between decimal and binary representations of positive integers be set up? Like this: 1 1 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 10 1010 11 1011 ... or like this: 1 1 2 3 4 5 6 7 8 9 10 10 11 11 ... -- Michael F. Stemper #include <Standard_Disclaimer> If we aren't supposed to eat animals, why are they made from meat? |