Week 2: Sets and Numbers II Flashcards
Numerals and Numbers
Numerals are symbols like 1, 2, or I, II, while numbers can be written using different numerals. ‘Two’ is the English name of a number, not a symbol.
Use and Mention
It’s important to distinguish between use and mention, e.g., a dog doesn’t contain any letters, but ‘Dog’ does.
Types and Tokens
Tokens are specific instances, e.g., the word ‘tattoo’ has 6 tokens of letters. Types are categories, e.g., ‘tattoo’ has 3 types of letters.
5 Types of numbers
Natural numbers: whole positive numbers; Integers: whole numbers, positive or negative; Rational numbers: any fraction of two integers where the denominator is not zero; Irrational numbers: cannot be expressed as a fraction of two integers; Real numbers: any number in the format 127.56 with the possibility of an infinite series of digits after the decimal point.
One-to-one correspondence
It’s a way of matching the members of set A with the members of set B so that every member of A is matched with one and only one member of B and vice versa.
Equinumerous
Two sets are equinumerous if they can be put into a one-to-one correspondence.
Smaller size of a set
A set A is smaller than a set B if A can be put into a one-to-one correspondence with a subset of B, but A cannot be put in one-to-one correspondence with B itself.
Infinite set
A set is infinite if it can be put in a one-to-one correspondence with one of its proper subsets.
Countable set
A set is countable if it is equinumerous to a subset of the set of natural numbers.
Uncountable set
A set is uncountable if it is infinite but cannot be put in a one-to-one correspondence with the set of natural numbers.
Uncountable infinities
The set of natural numbers is a smaller size of infinity than the set of real numbers.
Power set theorem
For any set S, the power set of S is not equinumerous with S.
Infinity of infinities
For every size of infinity, there is an even bigger size of infinity, i.e., the power set.
No set of all sets
There is no set of all sets because it would have to be a member of itself, creating the same problem as Russell’s paradox. Instead, proper classes are used to describe collections that cannot be considered sets.
