Where are we at?

Review of Cantor's Antinomy


Frege and Logicism: Frege background
Frege's Logicism
The Axioms
It is helpful to see the axioms of his system. Roughly, in contemporary formulation, and followed by both an english "translation" and also a very rough example, are the axioms. Let "A" stand for the universal quantifier, all. Let P, Q be sentences (these are each either true or false, but not both and not neither). Let F and G be predicates (these label properties). Let a be some particular thing.
Axiom 1 a and b:
a. (P → (Q → P))
b. (P → P)
"Translation": If sentence P is true, then if sentence Q is true, P is true.
Example: If the streets are wet, then if it rains then the streets are wet. It's not obvious at first why such a principle is useful, but it allows you to do many important things, like prove a conditional.

Axiom 2:
a. (∀xFx → Fa)
"Translation": If everything has property F, then some particular a has property F.
Example (assume we're talking only about numbers): If every number is divisible by 1, then 15 is divisible by 1.

b. (∀F(Fa) → Ga)
"Translation": If a has every property, then a has particular property G.

Axiom 3:
∀x∀y(x=y → ∀F(Fx → Fy))

"Translation": If x and y are the same, then any property x has is also had by y.
Example: If number x is the same as number y, then if x is divisible by 4, y is divisible by 4.

Axiom 4:
¬(P ↔ ¬Q) → (P ↔ Q)

"Translation": if it is not the case that sentence P is true just in case Q is false, then P is true just in case Q is true. Basically, this says that if it is not the case that P and Q have different truth values, then they have the same truth value.

Axiom 5
((x'Fx = x'Gx) ↔ ∀x(Fx ↔ Gx))

"Translation": the collection of things that F is true of is the same collection of things that G is true of, just in case for any thing x, x has property F if and only if x has property G. For example, the collections of humans is the same as the collection of rational animals, just in case if all and only humans are rational animals.

Axiom 6
a = ie'(a = e) "Translation": object a is the thing with property e such that a is the sole extension of e. This basically means that each object will have a unique description true only of it. Thus, suppose a stands for Abraham Lincoln. Then: a = the thing of the extension e where e is the property of being the 16th President of the United States and a is e.

The Problem
Frege needs Axiom 5 to develop the elements of mathematics that he aims to develop. However, he assumes that every concept is defined over every object. Axiom 5 also allows that the extension of a concept is an "object" -- it is a thing that we can quantify over and treat the way we treat other objects, like numbers, of our theory. Finally, Axiom 5 has as a consequence that once we formulate a concept it has an extension. These facts allow as concepts with extensions things like "the concept of being a concept that is not satisfied by itself."

The Paradox

I've quoted Frege from the Foundations of Arithmetic, translated by J. L. Austin, and quoted here out of The Frege Reader, Michael Beaney (ed.), 1997, Oxford: Blackwell publishers.