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A SET THEORETIC COSMOLOGY FOR

MATHEMATICS

BACKGROUND AND MOTIVATION

Nature of foundational work in mathematics

making sense of what we’re doing

giving a description of the domain

or universe we’re doing this in

finding a likely story, one that sticks

one without loose ends

Parallel examples

standard mathematics

nonstandard mathematics

Goal: a unified likely story for mathematics

based on sets and views of sets

Criteria: multiplicity + stability

THE COSMOLOGY: PRELIMINARIES

collections, membership 

sets: a, b, c, ….

Views: A, B, C, …..

Initial Axioms

Axiom 1: collections satisfy extensionality

Axiom 2: collections are closed under intersections

Axiom 3: for ever set a there exists a view A such

that a  A

Axiom 4: [preliminary version!] views are directed

by inclusion

INITIAL NOTIONS

 view A see s set a : a  A

 A’s version of a : the collection aA = a  A

and when A  A and A sees a

 an A-model of A’s (version of) a :

any a A for which aA = aA

 an A-model of A : any b A for which A = bA

in which case the model is transitive [small] if

A|= b is transitive [countable]

Axiom4: [2nd preliminary version!] for any views A, A there

exists view A, A  A having small models of A, A .
ITEMS FOR THE COFINAL CONSENSUS OF VIEWS :

ACCURACY

Types of Accuracy

 finiteness

 well-foundedness

 facts expressible by restricted formulas

[  -accuracy: to be dealt with later…]

Definition: View A is accurate for finiteness if in every case

A |= a is finite implies a  A .

Definition: View A is accurate for well-foundedness if whenever

A sees a, b for which

A |= b is a well-founded relation on a

then for each A  A there exists RF-accurate [defined below] view

A  A with models a, b of A’s a, b for which

A|= b is a well-founded relation on a .

DEFINITION OF RF-ACCURACY (MORE TECHNICAL)

Preliminary Notions

Godel Formulas : any of the form z = F(x, y) or

z  F(x, y) where F(x, y) is one of the

ten Godel operations

Godel Closed View : any view A such that for

each F(x, y) A |= ( x, y)( z)[z = F(x, y)]

Definition: View A is pre-Godel accurate if to each A  A there exists A  A so that the inclusion A  A is elementary for Godel formulas. View A is Godel accurate if to each A  A there exists pre-Godel accurate A  A for which the inclusion A  A is elementary for Godel formulas.

DEFINITION OF RF-ACCURACY (CONCLUSION)

Key Result: any inclusion A  A of Godel accurate views

is elementary for Godel formulas.

Corollary: any inclusion A  A of extensional, Godel closed,

Godel accurate views is elementary for restricted

formulas .

Definition: View A is accurate for restricted formulas

(RF-accurate) if to all A  A there exists extensional, Godel closed, Godel accurate A  A for which the inclusion A  A is elementary for restricted formulas .

AXIOM 4 COMPLETED

Definition: View A is accurate if it is accurate for finiteness,

well-foundedness and restricted formulas.

Definition: View A is modern if A |= GB-C where GB-C

is Godel Bernays set theory without foundation but with global

choice via an ordinal-indexed well ordering of all sets .

Axiom 4: [final version] For any views A, A there exists modern,

accurate A, A  A with small models of A, A . If

(say) A  A and A is RF-accurate, the A-model of

A can be also assumed to be transitive .

MULTIPLICITY IN THE COSMOLOGY

Given views A  A , the notions of

an A-speculation concerning (extensions of) A

those which are strictly realized in A

those which are equivalent

those realized in a view A  A

(whose realization is RF-accurate )

those which are plausible

Axiom 5: Any plausible speculation by an RF-accurate view

A concerning another RF-accurate view A  A has an

RF-accurate realization in some modern, accurate A  A .

A LAST ITEM FOR THE COFINAL CONSENSUS:

-ACCURACY

notation, certain standard ZFC formulas

Nat(x) : “x is a natural number”

k(x) : “x = the k-th natural number”

[k standard !]

Axiom 6: If A is any modern, accurate view then

A |= ( x)[Nat(x) Vkk(x)]

CONSEQUENCES OF THE AXIOMS

  • RF-accurate views are cofinal and their inclusions

are elementary for restricted formulas

  • For modern, accurate A  A with a  A the

following are equivalent:

A|= a is finite

A |= a is finite

a  A

a  A is a set

  • If modern, accurate A has a model of A  A , then

A can correctly determine A’s level of accuracy and

whether or not it is modern .

A MODEL OF THE COSMOLOGY

Theorem: Assume the following : ZFCB ,  -accuracy and, if

C is the global choice class , then there exists transitive set V0

such that

< V0 , |V0, V0 C > |= ZFCB

for any countable S  V0 , S  V0.

Then a model of the cosmology exists.

THE MODEL PARADOX !!!

Question: If this Cosmology is to be the grand theory of everything,

then how can we stand outside of it and even be looking at a

model of it ??

Answer: [a religious dogma to accompany the Cosmology] Nothing

exists outside of the Cosmology. What exists is what can be seen

by a view. We ourselves are within a modern, accurate view.

Our theory of the wider Cosmology that we’re a part of is a (self)

vision available to any view such as ours .

SELF-VISIONS AVAILABLE TO A Z-STRUCTURE M = <M, E>

-languages L

-fragments F

-theories T = (F, Ax)

M-focused -fragments : (F, L, name, ME)

the category M+

effective M-focused -fragments

self-visions available for M : (F, Ax, L, name, ME)

THE THEOREM OF COSMIC VISIONS

ASSUME M |= GB-C +  -accuracy

n is a natural number

x is a finite tuple of variables

THEN M has available self-vision (F, Ax, L, name, ME) where Ax  F consists of the following sentences :

all axioms of the Cosmology

modern-accurate-view(ME)

( x)[ x  ME VpM x = name(p) ]

and

ME |= ( name(p) )

for each finitary pure -formula ( x )  n(x) and x-ary tuple p of M-elements for which M |= ( p ) .