Form of all mathematical statements

[Mr Davis 97]

I am reading Real Mathematical Analysis by Pugh, and he claims that "All mathematical assertions take an implication form a --> b."
However, is this really true? For example, the assertion, "There exist infinitely many prime numbers," doesn't seem to take the if-then form.

 

[andrewkirk]

It is not true for axioms.

For example, of the Zermelo-Frankel axioms that are the foundation of set theory, one of them, the Null Set axiom ($\exists x\neg\exists y(y\in x)$) does not use an entailment symbol $\to$.

Another foundational example is the Peano axioms that found the natural numbers. The first one ($0\in\mathbb N$) contains no entailment.

However, nearly all theorems (as opposed to axioms) have the entailment form. They typically have a set of premises, and a conclusion that follows from the premises - therefore an entailment. For example, every continuous function on a compact set is bounded. This may be restated as 'IF f is a function that is continuous and has a compact domain THEN f is bounded'.

The statement about prime numbers is in entailment form, in the way it is usually proved and presented, which is:

'IF $x$ is prime THEN there exists an integer $y>x$ that is also prime'

The statement 'there are infinitely many primes' sounds like it contains no entailments, but when we interrogate the definitions of 'prime' and 'infinitely many' we will probably find that we need entailments to make those definitions. For instance, a common definition of $p$ being prime is:

$$
\forall x\forall y\ ((x\in\mathbb N\wedge y\in\mathbb N \wedge x\cdot y=p)\ \to\ (x=p\vee x=1))$$

However, an example of a theorem that does not use entailment is
1+1=2
which, in the language of Peano, is
$$S(0)+S(0)=S(S(0))$$

 

[mfb]

You can always force every true statement $a$ into that form by writing $true \Rightarrow a$.
$x=y$ can be written as $z=x \iff z=y$.
Possible? Yes. Useful? No.

 

[fresh_42]

I am reading Real Mathematical Analysis by Pugh, and he claims that "All mathematical assertions take an implication form a --> b."
However, is this really true? For example, the assertion, "There exist infinitely many prime numbers," doesn't seem to take the if-then form.

In general, mathematics is a deductive science: a conclusion of a truth from another truth, i.e. an implication. Physics, e.g. is a descriptive or inductive science.

Axioms are the framework, in which deductions take place. They describe the setting and are as such no implications, as gravitation is the setting and not the force. You have to start somewhere, if all is written as an implication. And these starting points are called axioms: assumed truths.

The example $\{true\} \longrightarrow A$ is the hidden case of axioms, because the truth we started at is either an axiom itself, or an earlier implication from a previous truth. It isn't all of a sudden just there.

"There exist infinitely many prime numbers" is an implication in other words. It really means the following:
If we consider numbers to be the elements that obey ...<arithmetic axioms> ... and call a number prime, if ... <definition of prime> ... then the set of natural numbers contains infinitely many of them.

Thus all three examples do not disprove Pugh, rather explain how it should be read. The kernel of his statement is true, as it was probably meant to describe the nature of mathematics, e.g. in contrast to physics, and not the difference between axioms and implications.

 

[FactChecker]

As the mathematical statements become more formal, they may tend to be expressable in the form a → b. Not all statements are formal and even the formal statements may be expressed in different ways.

The original example "There exist infinitely many prime numbers," can be expressed in the form a → b. :

If S is the set of prime numbers, then the cardinality of S is $\aleph$₀​