# Infinite Natural Numbers: First-Order Logic Formula Explained

• MHB
• pooj4
In summary, there are an infinite number of natural numbers and this can be proven by the following facts: (i) There is at least one natural number, (ii) Each natural number has a distinct successor, (iii) No two natural numbers have the same successor, and (iv) 0 is not the successor of any number. To create a formula in first-order logic that is only satisfiable by an infinite domain, one can use the following formula: (\forall x\forall y\,(S(x)=S(y)\to x=y))\land \forall x\,S(x)\ne0. This formula contains non-logical vocabulary and can only be satisfied if the domain is infinite.
pooj4
There are an infinite number of natural numbers. Why is that? Well this follows from the following facts:

(i) There is at least one natural number.

(ii) For each natural number there is a distinct number which is its successor, i.e., for each number $x$ there is a distinct number $y$ such that $y$ stands in the
successor relation to $x$.

(iii) No two natural numbers have the same successor.

(iv) There is a natural number, namely 0, that is not the successor of any number.Bearing these facts in mind, what's a formula in first-order logic that that is satisfiable by a valuation only if the domain of the valuation is infinite. Contain some non-logical vocabulary in presentation of course.

$$\displaystyle (\forall x\forall y\,(S(x)=S(y)\to x=y))\land \forall x\,S(x)\ne0$$

Evgeny.Makarov said:
$$\displaystyle (\forall x\forall y\,(S(x)=S(y)\to x=y))\land \forall x\,S(x)\ne0$$

thanks that helps

## 1. What are infinite natural numbers?

Infinite natural numbers are numbers that continue infinitely in a specific pattern, with each number being one more than the previous number. This set of numbers is denoted by the symbol ℕ and includes all positive whole numbers starting from 1.

## 2. What is first-order logic formula?

First-order logic formula, also known as first-order predicate calculus, is a formal system used in mathematics and computer science to express statements and arguments using logical symbols and quantifiers. It allows for the precise representation of mathematical concepts and reasoning.

## 3. How are infinite natural numbers expressed in first-order logic formula?

Infinite natural numbers can be expressed in first-order logic formula using the Peano axioms, which provide a set of rules for defining the natural numbers. These axioms include the successor function, which states that every natural number has a unique successor, and the induction axiom, which allows for the proof of statements about infinite natural numbers.

## 4. What is the significance of using first-order logic formula to explain infinite natural numbers?

Using first-order logic formula to explain infinite natural numbers allows for a precise and rigorous understanding of the concept. It also allows for the development of mathematical proofs and theorems about infinite natural numbers, which can have practical applications in various fields such as computer science and physics.

## 5. Are there any limitations to using first-order logic formula to explain infinite natural numbers?

While first-order logic formula is a powerful tool for understanding and reasoning about infinite natural numbers, it does have some limitations. For example, it cannot fully capture the concept of infinity, as it is a finite system. Additionally, it may not be able to express certain concepts or statements that require higher-order logic or other mathematical systems.

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