Unformatted text preview: on.
I.e., we have a consistency proof for mathematics (ZFC)
relative to that of T. 3
Furthermore, relative consistency proofs arising this way
are generally very finitary.
AMBITION:
Newton/Leibniz calculus:
Science and engineering.
=
Concept Calculus:
Everything else.
3. BETTER THAN, MUCH BETTER THAN.
We begin with the notions: better than (>), and much better
than (>>). These are binary relations. The passage of > to
>> is an example of what we call concept amplification.
One can also view > and >> mereologically, as
x > y iff y is a “proper part of x”. x >> y iff y is a “small proper part of x”.
Define z is minimal if and only if ÿ($w)(z > w).
Define z is minimally > y ´ ("w)(z > w ´ (y > w ⁄ y = w)).
The nonlogical axioms of T(>,>>,=)
BASIC. ÿx > x, x > y Ÿ y > z Æ x > z, x >> y Æ x > y. x >>
y Ÿ y > z Æ x >> z. x > y Ÿ y >> z Æ x >> z. ($x)(x >>
y,z). x >> y Æ ($z)(x >> z Ÿ z is minimally > y).
MINIMAL (Political Axiom). There is nothing that is better
than all minimal things.
EXISTENCE (Exact Bound, Plenitude). Let x be a thing better
than a given range of things. There is something that is
better than the given range of things and the things that
they are better than, and nothing else. Here we use
L(>,>>,=) to present the range of things.
Existence is like fusion. Here the “range of things” is
given by a first order formula in L(>,>>,=) with parameters
allowed. 4 AMPLIFICATION (Special Indiscernibility). Let y >
given, as well as a true statement about x, using
binary relations >,= and the unary relation >> x.
corresponding statement about x, using >,= and >>
also true. x be
the
The
y, is THEOREM 3.1. Basic + Minimal + Existence + Amplification is
mutually interpretable with ZFC. This is provable in EFA.
AMPLIFIED LIMIT (Star). There is something that is better
than something, and also much better than everything it is
better than.
BINARY AMPLIFICATION. Let y > x be given, as well as a true
statement about x, using the binary relations >,= and the
binary relation z >> w >> x. The corresponding statement
about x, using >,= and z >> w >> y, is also true.
Leads to much higher places in Interpretation Hierarchy
than ZFC:
THEOREM 3.2. Basic + Minimal + Existence + Amplification +
Amplified Limit interprets ZFC + “there is an almost
ineffable cardinal” and is interpretable in ZFC + “there
exists an ineffable cardinal”.
THEOREM 3.3. Basic + Minimal + Existence + Binary
Amplification interprets ZFC + “there exists a Ramsey
cardinal” and is interpretable in ZFC + “there exists a
measurable cardinal”.
THEOREM 3.4. Basic + Minimal + Existence + Binary
Amplification + Amplified Limit interprets ZFC + “there
exists a measurable cardinal with arbitrarily large lesser
measurable cardinals” and is interpretable in ZFC + “there
exists a measurabl...
View
Full
Document
This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Calculus, Set Theory

Click to edit the document details