fom
Posts:
1,968
Registered:
12/4/12


Re: ZFC is inconsistent
Posted:
Mar 17, 2013 5:54 AM


On 3/16/2013 4:57 PM, WM wrote: > On 16 Mrz., 16:13, fom <fomJ...@nyms.net> wrote: > >> Where we come to the question of >> how you refer to points without >> an implicit use of infinity. > > All points that you can define geometrically, belong to a finite > collection. >> >> This, of course, comes back to >> how you refer singularly without >> an implicit use of infinity. > > A unit can be defined without referring to infinity. I think it is one > of the silliest arguments of matheology that infinity is required to > define finity. It is simply insane.
No. It is merely respectful  something of which you seem incapable.
============================================
"A point is that which has no part."
Euclid
"For when we spoke of things in a subject, we did not mean things belonging in something as parts"
Aristotle
"What St. Thomas affirms on this point about angels or intelligences ('that here every individual is a lowest species') is true of all substances, provided one takes the specific difference in the way that geometers take it with regard to their figures."
Leibniz
"If m_1, m_2, ..., m_v, ... is any countable infinite set of elements of [the linear point manifold] M of such a nature that [for closed intervals given by a positive distance]:
lim [m_(v+u), m_v] = 0 for v=oo
then there is always one and only one element m of M such that
lim [m_(v+u), m_v] = 0 for v=oo"
Cantor to Dedekind
"If x is any object of the domain, there exists a set {x} containing x and only x as element"
"The question whether x=y or not is always definite since it is equivalent to the question whether or not xe{y}"
Zermelo
"A unit is that by virtue of which each of the things that exist is called one"
Euclid
"The more I have thought the matter over, the more convinced I have become that arithmetic and geometry have developed on the same basis  a geometrical one in fact  so that mathematics in its entirety is really geometry"
Frege
"In whatever manner and by whatever means a mode of knowledge may relate to objects, intuition is that through which it is in immediate relation to them, and to which all thought as a means is directed."
"Objects are given to us by means of sensibility, and it alone yields us intuitions"
"By means of outer sense, a property of our mind, we represent to ourselves objects as outside us, and all without exception in space"
"Geometry is a science which determines the properties of space synthetically, and yet a priori."
Kant
"..., I shall deal first with projective geometry. This, I shall maintain, is necessarily true of any form of externality, and is, since some such form is necessary to experience, completely a priori."
"We can distinguish different parts of space, but all parts are qualitatively similar, and are distinguished only by the immediate fact that they lie outside one another"
"Analysis, being unable to find any earlier haltingplace, finds its elements in points, that is, in zero quanta of space"
"A point must be spatial, otherwise it would not fulfill the function of a spatial element; but again, it must contain no space, for any finite extension is capable of further analysis. Points can never be given in intuition, which has no concern for the infinitesimal."
Russell
A1: If P is a part of object Q, then Q is not a part of object P
A2: If P is a part of object Q, and Q is a part of object R, then P is a part of object R
D1: P is an ingredient of an object Q when and only when, P is the same object as Q or is a part of object Q
D2: P is the class of objects p, when and only when the following conditions are fulfilled:
a) P is an object
b) every p is an ingredient of object P
c) for any Q, if Q is an ingredient of object P, then some ingredient of object Q is an ingredient of some p
Lesniewski
And, to put Lesniewskian ideas into a form that is related to ZF and compatible with the deductive calculus of firstorder predicate logic,
news://news.giganews.com:119/VoadnTfTyd4bgt3MnZ2dnUVZ_vOdnZ2d@giganews.com
with model theoretic considerations
news://news.giganews.com:119/5bidnemPpsnq13zNnZ2dnUVZ_sOdnZ2d@giganews.com
and a construction of Dedekind cuts
news://news.giganews.com:119/M5qdncGgG5aVbvMnZ2dnUVZ_rCdnZ2d@giganews.com
as well as a geometric basis for the logical language
news://news.giganews.com:119/Jr2dnbdYvtfPdlrNnZ2dnUVZ_tdnZ2d@giganews.com
news://news.giganews.com:119/IqudndogJ8VB1zNnZ2dnUVZ_qydnZ2d@giganews.com
news://news.giganews.com:119/zsCdnW9U7v4BOlzNnZ2dnUVZ_hmdnZ2d@giganews.com
news://news.giganews.com:119/AuqdnYcXm8eaLVzNnZ2dnUVZ_hdnZ2d@giganews.com
news://news.giganews.com:119/EsqdnX0_NvwwKlzNnZ2dnUVZ_odnZ2d@giganews.com
news://news.giganews.com:119/Jr2dnbZYvtc2dlrNnZ2dnUVZ_tdnZ2d@giganews.com
news://news.giganews.com:119/dbydnS9OZYESWFzNnZ2dnUVZ_uidnZ2d@giganews.com
news://news.giganews.com:119/Jr2dnbFYvtcHcVrNnZ2dnUVZ_tdnZ2d@giganews.com
news://news.giganews.com:119/Jr2dnbBYvtedcFrNnZ2dnUVZ_tdnZ2d@giganews.com
news://news.giganews.com:119/Jr2dnbNYvtf9cFrNnZ2dnUVZ_tdnZ2d@giganews.com
news://news.giganews.com:119/Jr2dnbNYvtf9cFrNnZ2dnUVZ_tdnZ2d@giganews.com

