Axiomata

Frege

Law III: Identity of Indiscernibles

If two objects have the same properties, they are the same object.

\[\forall a, b: (a = b) \equiv (\forall f: f(a) \equiv f(b))\]

Law V: Extensionality

\[(\hat{\epsilon}f = \hat{\epsilon}g) \equiv (\forall x: f(x) \equiv g(x))\]

Peano

Axioms of Arithmetic

1. \(1 \in \mathbb{N}\)

2. \(a \in \mathbb{N} \supset a = a\)

3. \(a,b \in \mathbb{N} \supset (a = b) = (b = a)\)

4. \(a, b, c \in \mathbb{N} \supset (a = b) \land (b = c) \land (a = c)\)

5. \((a = b) \land b \in \mathbb{N} \supset a \in \mathbb{N}\)

6. \(a \in \mathbb{N} \supset (a + 1) \in \mathbb{N}\)

7. \(a,b \in \mathbb{N} \supset (a = b) \land (a + 1) \land (b + 1)\)

8. \(a \in \mathbb{N} \supset a + 1 \neq 1\)

9. \((((k \in K) \land (1 \in k) \land (x \in \mathbb{N}) \land (x \in k)) \supset_x (x+1) \in k) \supset (N \supset k)\)

Kolmogorov

Axiom of Continuity

If \(A_i\) is a sequence of decreasing events,

\[A_1 \supset A_2 \supset ... \supset A_n\]

And if the intersection of those events is empty,

\[\cap_{i=1}^n A_i = \varnothing\]

Then the limit as of the probability of this sequence is zero,

\[\lim_{n \to \infty} P(A_n) = 0\]

Tarski

Deductive Science

Definitions

1. \(S\): The set of all meaningful sentences in a science.

2. \(A\): An arbitrary subset of \(S\).

3. \(C_n(A)\)

4. \(E_{f(x)}[ ... ]\): The set of all values of the function f corresponding to those values of the argument x which satisfy the condition formulated in the brackets “[..]”.

5. \(\mathbb{P}(A) = E_X[X \subseteq A]\): The powerset of A, i.e. the set of all subsets of A.

6. \(\mathbb{F} = E_X[ \lvert X \rvert \leq \aleph_0]\): The set of all finite “inductive”sets.

Axioms of Deductive Science

1. The number of sentences is finite or infinite.

\[\lvert S \rvert \leq \aleph_0\]

2. If \(A \subseteq S\) then \(A \subseteq C_n(A) \subseteq S\)

3. If \(A \subseteq S\) then \(C_n(C_n(A)) = C_n(A)\)

4. If \(A \subseteq S\) then \(C_n (A) = \sum_{X \in \mathbb{P}(A) \cdot \mathbb{F}} C_n(X)\)

Mereology

Definition: Length of Open Interval

The length \(l(I)\) of an open interval \(I\) is defined by,

1. If \(I = (a,b)\) for some \(a, b \in \mathbb{R}\) with \(a < b\), then \(l(I) = b - a\)

2. If \(I = \emptyset\), then \(l(I) = 0\).

3. If \(I = (-\infty, a)\) or \(I = (a, \infty)\),for some \(a \in \mathbb{R}\), then \(l(I) = \infty\).

4. If \(I = (-\infty, \infty)\), then \(l(I) = \infty\)

Definition: Outer Measure

The outer measure \(\lvert A \rvert\) of a set \(A \subset \mathbb{R}\) is defined by,

\[\lvert A \rvert = \text{inf}\{ \sum_{k = 1}^{\infty} l(I_k) : I_1, I_2, ... \text{are open intervals such that} A \subset \cup_{k = 1}^{\infty} I_k \}\]

Definition: Parts

\(\subset\) (part-of relation): A binary relation that holds between two individuals when one is a part of the other.

Definition: Disjoint

\(\otimes\) (disjoint relation): A binary relation that holds between two individuals when they have no common parts.

Definition Sum

Every element of α is a part of a sum,

\[\forall y: y \in \alpha \to y \subset \sum \alpha\]

Reflexivity

Every individual is a part of itself.

\[\forall x: x \subset x\]

Transivity

If x is a part of y, and y is a part of z, then x is a part of z.

\[\forall x: \forall y: \forall z: ((x \subset y) \land (y \subset z)) \to (x \subset z)\]

Antisymmetry

If x is a part of y, and y is a part of x, then x and y are identical.

\[\forall x: \forall y: ((x \subset y) \land (y \subset x) \to x = y)\]

Supplementation

If x is not a part of y, then there exists a part z of x that is disjoint from y

\[\forall x: \forall y: \neg(x \subset y) \to ((\exists z: z \subset x) \land (z \otimes y))\]

Summation

For any non-empty class α of individuals, there exists an individual x that is the sum of all elements of α.

\[\forall \alpha \forall x: x = \sum \alpha\]

Atomicity

Every non-empty class of individuals has an element that shares no parts with any other element.

\[\forall \alpha: \alpha \neq \emptyset \to (\exists x: (x \in \alpha) \land (\neg \exists y:(y \in \alpha) \land (y \subset x) ))\]

Zalta

Definitions

• Ox: x is an ordinary individual

• Ax: x is an abstract individual

• xF: x encodes the property F

• Fx: x exemplifies the property F

Abstract Objects

1. Ordinary individuals necessarily and always fail to encode properties.

\[\forall x: Ox \to \Box \neg \exists F: xF\]

2. For every condition on properties, it is necessarily and always the case that there is an abstract individual that encodes just the properties satisfying the condition.

\[\forall \phi: \exists x: Ax \land \Box \forall F: (xF \leftrightarrow \phi(F))\]

3. Two individuals are identical if and only if they are both ordinary individuals and they necessarily and always exemplify the same properties, or they are both abstract individuals and they necessarily and always encode the same properties.

\[\forall x: \forall y: (x =y) \leftrightarrow [ (Ox \land Oy \land \Box \forall F: (Fx \leftrightarrow Fy)) \lor (Ax \land Ay \land \Box \forall F: (xF \leftrightarrow yF))]\]

4. If it is possible or sometimes the case that an individual encodes a property, then that individual encodes that property necessarily and always.

\[\forall x: \forall F: \Diamond xF \to \Box xF\]

5. For every exemplification condition on individuals that does not involve quantification over relations, there is a property which is such that, necessarily and always, all and only the individuals satisfying the condition exemplify it.

\[\forall \phi: \exists F: \Box \forall x: Fx \leftrightarrow \phi(x)\]

6. Two properties are identical just in case it is necessarily and always the case that they are encoded by the same individuals.

\[\forall F: \forall G: (F = G) \leftrightarrow \Box \forall x: (xF \leftrightarrow xG)\]