Philosophical Devices 2. Infinite Sets

What are integers?

"If we add the negative whole numbers to the natural numbers, then we get the integers: ... -3, -2, -1, 0, 1, 2, 3 ...

How many words were there in: cat cat?

One word type, but two tokens of that type.

Can real numbers be paired with natural numbers?

The real numbers cannot be paired one-to-one with the natural numbers. They are non-denumerable. Indeed the reals between 0 and 1, or in any finite interval, are non-denumerable.

What are real numbers?

The real numbers comprise both the rational and irrational numbers. Any real number can be represented by an infinitely long decimal expansion: e.g. 23.17564839 ...

In what sense can a subset B be paired with the set A?

If a finite set B is a proper subset of finite set A, then the As can't all be paired up one-to-one with the Bs, for there won't be enough Bs— any attempted pairing will leave some extra As unpaired. But with infinite sets, B can be a proper subset of A, and still be paired up one-to-one with the As— for now the Bs won't automatically run out before we get to the end of the As.

What is one way of distinguishing words from the things that they are referring​ to?

If you want to talk about the word rather than the thing, you must put the word in quotes to form a name of that word itself. Ex. London contains ten million people, but 'London' contains six letters.

Are there more natural numbers than even numbers?

In one obvious sense the answer must be yes. The set of even numbers {0, 2, 4, 6, ...} is a proper subset of the set of natural numbers {0, 1, 2, 3, ...}. The latter set contains all the members of the former set and then some. There are plenty of natural numbers that aren't even, but no even numbers that aren't natural. This is the subset sense. But in the pairing sense no.

What does Cantor's famous diagonal argument show?

It shows that there are more real numbers than natural numbers even in the one-to-one pairing sense of 'more'. If you try to pair up the reals with the naturals you will always have some real number left over. Given any supposed listing of the reals, it is always possible to construct another real number that isn't in that list.

What is a numerical list?

A numerical list, if you think about it, just is a pairing of the listed items with the natural numbers— the first in the list with 1, the second with 2, and so on.

What is the distinction between numbers and numerals?

Numbers are the more abstract things that these numerals name. The English word 'two' is a different word from the French word 'deux', but they both name the same number.Numbers themselves are timeless entities that transcend the perspective of any given system of representation.

What are numerals?

Numerals are words like 'one' and 'two' or symbols like '1' and '2'. Numerals are signs used in specific representation systems.

In what sense are there just as many natural numbers as even numbers?

The even numbers can be paired up one-to-one with the naturals. In this sense there are just as many even numbers as natural numbers.This is the pairing sense.

What is the defining characteristic of infinite sets?

The members of any infinite set, but of no finite set, can be paired up one-to-one with the members of some of its proper subsets.

What are the most basic numbers?

The most basic numbers are the natural numbers: 0, 1, 2, 3, ...

What are rational numbers?

They can be expressed as fractions of the form p/ q, where p and q are integers.

What does it mean to be denumerable?

Whenever the members of a set can be paired one-to-one with the natural numbers, we say the set is denumerable. A denumerable set is one that can be placed in a numerical list.

What is the mathematical property of density?

given any two rational numbers, however close together, there will always be another rational number in between them.

What is the set of all natural numbers?

{0, 1, 2, 3, ...}.