Russels Revenge

ok , lets give an overview of the " foundations of math "

i could have said ' set theory ' but then people would say there are alternatives to sets e.g. classes , that ' set theory ' is not correct etc etc

( axiomatic set theory might be accepted better )

as the title said : Russels Revenge.

although it is ofcourse my own revenge , in the spirit of Russel.

no name calling , just an overview of the foundations , axiomatic systems or whatever you want to call them.

objective , but with skeptical remarks.

keeping in mind of course that TST is not discussed and is thus the alternative.

since the discussed foundations are the most common , it is clear that TST is probably the only alternative , or at least other good alternatives are similar or based upon TST.

you will notice that there is an order in the proposed foundations , as by improvement and generality or at least the attempt of it.

we start :


1) naive set theory

indeed naive :

a) not axiomatic

b) russels paradox : the set of all sets

2) ZFC

( im not mentioning cantor )

ok , ZFC was ' suppose to ' resolve russels paradox.

BUT !! it doesnt !!

it simply FORBIDS russels paradox as INVALID

similar to a theory that explains complex numbers by FORBIDDING sqrt(-1) !!

so , in a way russels paradox is still there !!

this leads to a delusional solution called " proper class "

and now we arrive at the final destruction of ZFC !

a " proper class " does not obey the laws of ZFC !!

the axiom of seperation only tells that the set of all sets cannot be a set.

still sounds like a paradox hmm

the set that cant be a set.

but lets ignore that and settle with proper class.

proper classes are outside of ZFC.

they do not obey ZFC since ZFC are axioms about sets and a proper class is not a set ( ZFC says itself ! )



so here is the solution to russel and its axiomatic system

russel -> zfc -> proper classes -> ??

we thus see that ZFC DID NOT solve russels paradox and is therefore as inconsistant as naive set theory ,

Unless we find an axiomatic system for proper classes , THAT IS ALSO consistant with ZFC.

so lets try to find them , and keep in mind that if we cant russel is still a paradox and ZFC is both incomplete(e.g. proper classes) and inconsistant (e.g. russel).

( and ZFC is not provable either , as expected from axioms both inconsistant and incomplete ! )

we seek and find the following for both sets and proper classes ( consistant with ZFC ) :

3) NBG

so NBG is supposed to solve the issue of russel and be a foundation for proper classes.

but quite rapidly we notice the following :

NBG accepts " the class of all sets "

BUT NOT

" the set of all sets "

NOR

" the class of all classes "

one could say the set of all sets is replaced by the class of all sets.

but there is no excuse for

" the class of all classes "

so to avoid russel , we called some 'weird sets' classes.

but russel remains : " the class of all classes " !!!


not consistant with NGB !!

so what i wrote about NGB applies to ZFC as well !

and NGB ' consistancy and completeness was needed for that of ZFC.

but as the class of all classes shows , NGB isnt complete nor consistant , thus neither is ZFC.

and russel is still unresolved.

way out ?

searching for yet another foundation for ZFC or a foundation for the generalization of proper classes ( NGB )

4) MK ( morse-kelley )

quite similar to NGB and no extension of proper classes.

same domain of discourse thus.

no solution.

nor a new start since related to NGB and ZFC in a canonical way.

5) NF

the impossible Russel class is not an NF set

a new paradox occurs see :

Set Theory over Classes. (Dissertation). Kiel 1973

by Arnold Oberschelp.

6) Fuzzy Set theory

the fuzzy is indeed very fuzzy.

there is ' degree of membership ' !?

doesnt resolve russel either.

doesnt match with ZF.

7) NFU

NFU is not consistant with the axiom of choice.

thus not consistant with ZFC.

8) finite set theory

of course no sets with infinite members.

not a solution either and very limited.

9) semiset

fuzzier than fuzzy set theory

more paradoxal than russel

vague properties

10) positive set theory

doesnt work out.

strongly related to the above mentioned.

( axiom of specification )

11) internal set theory

confusing set theory with infinitesimals and such.

12) non-well founded set theory

?

13) leaving out regularity

doesnt solve russels paradox

( and 13) is just a step , not a complete axiomatic system , and not really independant of the above 12 )


14) urelements

independant ; one can take ZF + urelements

NF + Urelements = NFU

both already discussed.

since independant not a solution per se...

... nor an axiomatic set theory.


15) working with ordinals

doesnt help ; russels " set of all sets "

basicly becomes " set of all ordinals "

16) large cardinals and ' universes '

! wow ! , with what axiomatic set theory ?

assumes russel is ALREADY (?!) solved and its solution ( whatever you call it ) has a big cardinality ( based on nothing thus ! )


AND THATS THE END OF THE LIST OF THE MOST POPULAR !!!


Russels revenge haha

russels paradox still lives !!

also note godel's theorems are only valid in ZF !

and this is the foundation of math ??!!??

oh my !!

and i didnt even mention other set theory paradoxes , only russel !

id rather bet on calculus as a foundation.

foundations : 1/10


regards

tommy1729
.

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