Four basic concepts

Another word, which famously has three quite distinct meanings, is “is”. The three meanings are illustrated in the following three sentences.

(1) 5 is the square root of 25.
(2) 5 is less than 10.
(3) 5 is a prime number.

In the first of these sentences, “is” could be replaced by “equals”: it says that two objects, 5 and the square root of 25, are in fact one and the same object, just as it does in the English sentence “London is the capital of the United Kingdom.”

In the second sentence, “is” plays a completely different role. The words “less than 10” form an adjectival phrase, specifying a property that numbers may or may not have, and “is” in this sentence is like “is” in the English sentence “grass is green.” As for the third sentence, the word “is” there means “is an example of”, as it does in the English sentence “Mercury is a planet.”

These differences are reflected in the fact that the sentences cease to resemble each other when they are written in a more symbolic way. An obvious way to write :

(1) is 5 = p25. As for
(2) it would usually be written 5 < 10, where the symbol < means “is less than”.

The third sentence would normally not be written symbolically because the concept of a prime number is not quite basic enough to have universally recognised symbols associated with it. However, it is sometimes useful to do so, and then one must invent a suitable symbol. One way to do it would be to adopt the convention that if n is a positive integer, then P(n) stands for the sentence “n is prime”.

Another way, which doesn’t hide the word “is”, is to use the language of sets.