# Proving Primitive Symbols with Axioms

• MHB
• solakis1
In summary, we use the given axioms to show that 1 is not equal to any other primitive symbol and then use a proof by contradiction to show that for any primitive symbol x, either x is equal to 1 or there exists some primitive symbol y such that y* = x.
solakis1
Given :

A) primitive symbols : (1, *) and

B) The axioms:

1) $$\displaystyle \forall x\forall y[x*=y*\Longrightarrow x=y]$$

2) $$\displaystyle \forall x[x*\neq 1]$$

3) $$\displaystyle [P(1)\wedge\forall x(P(x)\Longrightarrow P(x*))]\Longrightarrow\forall xP(x)$$

Then prove:

$$\displaystyle \forall x[x=1\vee \exists y(y*=x)]$$

First, we will use axiom 2 to show that 1 is not equal to any other primitive symbol. Assume the opposite, that 1 is equal to some primitive symbol x. By axiom 1, this means that 1* = x*. But this contradicts axiom 2, since 1* = 1 and we know that 1 ≠ 1. Therefore, 1 is not equal to any other primitive symbol.

Next, we will prove the statement \forall x[x=1\vee \exists y(y*=x)] by contradiction. Assume the opposite, that there exists some x such that x ≠ 1 and for all y, y* ≠ x. This means that there is no primitive symbol that can be multiplied by another primitive symbol to get x. This contradicts axiom 3, since it states that if P(1) is true and for all x, P(x) implies P(x*), then P(x) is true for all x. In this case, P(1) is true (since 1 ≠ 1 is false) and for all x, x ≠ 1 implies x* ≠ x (since we assumed this to be true). Therefore, by axiom 3, \forall x[x=1\vee \exists y(y*=x)] must be true.

In conclusion, by using the given axioms, we have proven that \forall x[x=1\vee \exists y(y*=x)]. This means that for any primitive symbol x, either x is equal to 1 or there exists some primitive symbol y such that y* = x.

## 1. What is the purpose of proving primitive symbols with axioms?

The purpose of proving primitive symbols with axioms is to establish a logical foundation for a mathematical system. By defining the primitive symbols and providing axioms, we can then use deductive reasoning to prove theorems and statements within that system.

## 2. What are primitive symbols?

Primitive symbols are the basic building blocks of a mathematical system. They are typically undefined and cannot be broken down into simpler components. Examples of primitive symbols include numbers, variables, and logical connectives.

## 3. What is an axiom?

An axiom is a statement that is accepted as true without proof. It serves as a starting point for logical reasoning and is used to deduce other statements within a mathematical system.

## 4. How do you prove primitive symbols with axioms?

To prove primitive symbols with axioms, we use deductive reasoning. This involves starting with the given axioms and using logical rules to derive new statements. By following this process, we can establish the truth of the primitive symbols within the defined system.

## 5. Are there different ways to prove primitive symbols with axioms?

Yes, there are different approaches to proving primitive symbols with axioms. Some methods include using truth tables, natural deduction, and formal proof systems. The choice of method may depend on the specific mathematical system and the complexity of the primitive symbols and axioms involved.

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