Logic — math, philosophy & computational aspects

Here is an example of diagonalization

123
456
789

Diag = 159

AntiDiag = 260   <<<<<<<NEW SEQUENCE NOT ON THE LIST!

YOU ALL THINK THIS WORKS ON THE LIST OF COMPUTABLE REALS!

DON’T YOU!!!

Gee it works for 159, must work in the infinite case too, who cares if there’s
no new digit sequence that can be formed.

You’re all DIM!  How can you form a new digit sequence when they’re all
computed up to infinite length?  

Or as George Greene puts it, they’re all computed up to ALL (infinite) FINITE lengths.

And as George Greene puts it there’s a new digit sequence at some FINITE point.

Well I can’t see it.

Herc
—
the nonexistence of a box that contains the numbers of all the boxes
that don’t contain their own box number implies higher infinities.
– Cantor’s Proof (the holy grail of paradise in mathematics)

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The voting paradox …is a situation noted by the Marquis de Condorcet
in the late 18th century, in which collective preferences can be
cyclic …even if the preferences of individual voters are not. This
is paradoxical, because it means that majority wishes can be in
conflict with each other. When this occurs, it is because the
conflicting majorities are each made up of different groups of
individuals. For example, suppose we have three candidates, A, B and
C, and that there are three voters with preferences as follows
(candidates being listed in decreasing order of preference):

Voter 1: A B C
Voter 2: B C A
Voter 3: C A B

If C is chosen as the winner, it can be argued that B should win
instead, since two voters (1 and 2) prefer B to C and only one voter
(3) prefers C to B. However, by the same argument A is preferred to B,
and C is preferred to A, by a margin of two to one on each occasion.
The requirement of majority rule then provides no clear winner.

Also, if an election were held with the above three voters as the only
participants, nobody would win under majority rule, as it would result
in a three way tie with each candidate getting one vote. However,
Condorcet’s paradox illustrates that the person who can reduce
alternatives can essentially guide the election. For example, if Voter
1 and Voter 2 choose their preferred candidates (A and B
respectively), and if Voter 3 was willing to drop his vote for C, then
Voter 3 can choose between either A or B – and become the agenda-
setter.

When a Condorcet method is used to determine an election, a voting
paradox among the ballots can mean that the election has no Condorcet
winner. The several variants of the Condorcet method differ on how
they resolve such ambiguities when they arise to determine a winner.
Note that there is no fair and deterministic resolution to this
trivial example because each candidate is in an exactly symmetrical
situation.

The phrase "Voter’s Paradox" is sometimes used for the rational choice
theory prediction that voter turnout should be 0.

http://en.wikipedia.org/wiki/Voting_paradox
http://en.wikipedia.org/wiki/Voting_system#Foundations_of_voting_theory
http://www.google.com/search?hl=en&q=define%3ATransitivity

Hi,

Could anyone help me with this question (from Bostock’s Intermediate
Logic)

"The theory of groups can be presented as having in its vocabulary
just identity and a single two-place function f(x,y) which we write as
‘x.y’. The usual laws for identity apply, and in addition these three
axioms:"

(A1) Axyz(x.(y.z) = (x.y).z)

(A2) AxyEz(x = z.y)

(A3) AxyEz(x = y.z)

Prove:

a = a.c   |=   c = c.c

(Hint: Use Ez(c = z.a))

Thanks for any help,
Mitch.

This is another way of writing FOL(membership and identity)
using a dot system, but not for punctuation.
Punctuation here is done by using the bracket system on need,
i.e. only when we have unexpected order of connectives.

Logical connectives in Ascending order of power:

Negation                 ~
Conjunction              ,
Disjunction               ;
Implication              ->
Bi-conditional         <->

Primitives

Membership       juxtaposing
Identity                   =

Quantifiers

Universal                :
Existential              .
Unique Existential   !

Syntax of quantification:

Quantifier x_Q quantifier

There is no need for the underscore if Q begins with a quantifier.

Hiding quantifiers:
 Quantifiers can be hidden if their appearance would be
consecutive otherwise; or in case of universal quantification
the detail of which is totally decidable without their appearance.

Brackets ( ): used in n-arity functions and predicates; for
punctuation: ONLY used to delineate *unexpected* order
of connectives.

Examples:

:t.x:yt<->:w.k:uw<->Wiener pair(u),:i.s.r_isru..->i subset k:,
                              :j.p.q_jpqu,0q..->j=x::.->yw::.:

for all t Exist x for all y ( y e t <-> for all w
( Exist k for all u ( u e w <->
(u is a wiener ordered pair &
for all i (Exist sr (iesereu) -> i subset k)&
for all j (Exist pq (jepeqeu & 0eq) -> j=x)))
-> yew)).

Extensionality  :x:y:zx<->zy:->x=y::

Foundation  :x.yx.->.zx,~.cz,cx..:

Empty  .x:y_~yx:.

Pairing   :a:b.x:yx<->y=a;y=b:.::

Union  :a.x:yx<->.z_yza.:.:

Power  :a.x:yx<->:zy->za::.:

Separation :a.x:yx<->ya,Q:.:

Infinity  .x_0x,:yx->:z:uz<->uy;u=y:->zx::.

Zuhair

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