details re Z-R |- PA consistent
in [Math]
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From: Marshall on 14 Jul 2010 02:49 On Jul 12, 6:04 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > If two posters make symmetrical yet opposing claims, then > absent any supporting information I see no reason to favor one > claim over the other. None of the cases you've ever fussed about could even remotely be described as "absent any supporting information." Marshall
From: Tim Little on 14 Jul 2010 03:44 On 2010-07-14, Marshall <marshall.spight(a)gmail.com> wrote: > On Jul 12, 6:04 pm, Transfer Principle <lwal...(a)lausd.net> wrote: >> If two posters make symmetrical yet opposing claims, then >> absent any supporting information I see no reason to favor one >> claim over the other. > > None of the cases you've ever fussed about could even > remotely be described as "absent any supporting information." Not entirely true. They could be considered "absent any supporting information" to someone who is too dense to comprehend any of the supporting information. Of course in all those cases a sufficient amount of information has been comprehensible to a 10-year-old, but in the case of certain readers that may be too much to expect. - Tim
From: FredJeffries on 14 Jul 2010 11:51 On Jul 12, 6:04 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > But we still might be able to salvage this theory, borrowing ideas > from NBG or NFU. In NFU (where V is a _set_ and Frege's > naturals are used instead of von Neumann's), the Axiom of Infinity > can be stated as: > > {} is not a natural number. > > For using Frege naturals, the natural number n is the set of all sets > of cardinality n. Since {} is not a natural number, it means that for > every n, the set of all sets of cardinality n is not {} -- i.e., there > is a > set (a subset of V) of every finite cardinality n -- therefore the > universe V must be infinite. > > In this case, we can write Herc's Axiom of No Infinity as: > > {} _is_ a (Frege) natural number. > > This forces the universe to be finite Why? Suppose (for some bizarre reason) there are no sets of cardinality 43. Why, despite this, could there not be sets of all cardinalities greater than 43? Or could it not be that there only exist sets of even cardinality and none of odd cardinality? Or that there are no sets with cardinality a prime number?
From: FredJeffries on 14 Jul 2010 12:01 On Jul 12, 4:57 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 11, 9:55 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > I am curious. Exactly what biases have I overcome? (This is an honest > > question: Since they are my own biases it is difficult for me to see > > them on my own.) > > I assume that Jeffries believes in infinite sets Mark Twain was once asked if he believed in infant baptism, to which he replied "Believe in it?! Why, I've seen it done!" > (i.e., he regularly > works in theories which prove their existence). Nope, I work in a cubicle. In the past I have worked in chicken houses, bookstores, factories, loading docks, ... but no theories on my resume. > So this represents > a bias towards infinite sets. Elsewhere you state there is nothing infinite in the real world. So doesn't everyone have a natural bias towards finitism? Therefore by your principle shouldn't anyone who is able to overcome this naive finitism and intelligently discuss infinite sets be supported and commended rather than those who have hitherto been unable to overcome their finitistic bias? But thank you for answering my question.
From: Marshall on 14 Jul 2010 12:38
On Jul 14, 12:44 am, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-14, Marshall <marshall.spi...(a)gmail.com> wrote: > > > On Jul 12, 6:04 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > >> If two posters make symmetrical yet opposing claims, then > >> absent any supporting information I see no reason to favor one > >> claim over the other. > > > None of the cases you've ever fussed about could even > > remotely be described as "absent any supporting information." > > Not entirely true. They could be considered "absent any supporting > information" to someone who is too dense to comprehend any of the > supporting information. Of course in all those cases a sufficient > amount of information has been comprehensible to a 10-year-old, but in > the case of certain readers that may be too much to expect. Just so! BTW, my working hypothesis is that TP is 12-14 years old. Marshall |