Philosophical Devices
Week 2 – Infinite Sets & Etc.
--------------------------------------------------------------------------------------------------------ABOUT RELATIONS
 Reflexivity: R is reflexive iff for all x in the domain, xRx
 Nonreflexive: at least one x fails to bear R to itself
 Irreflexive: for no x in domain, xRx
‘Strong’ and ‘weak’ negative properties. ‘Irreflexive’ is a strong negative - NO pairs <x,x>
are in the relation. ‘Nonreflexive’ is a weaker negative property, requiring only that there is at
least one pair <x,x> not in the relation. The non- form always simply means ‘not’, and the
stronger negation is always expressed with a Latin prefix: ir-, a-, in-.
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Symmetry: R is symmetric iff For all x & y in the domain, if xRy then yRx
Nonsymmetric: There is at least one pair <x,y> where xRy but not yRx
Asymmetric: For no pair <x,y>, if xRy then yRx
Antisymmetric: If xRy and yRx, then x = y
Note: an asymmetric relation must be irreflexive. When we consider ordered pairs, we have
to include cases where the variables x and y get the same value. Distinct variables can happen
to get the same value, and to consider all possibilities, you have to remember to include
those. However, an antisymmetric relation need not be reflexive!
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Transitive: R is transitive iff For all x, y, & z in the domain, if xRy and yRz, then xRz
Nontransitive: for at least one trio x,y,z, xRy and yRz, but not xRz
Intransitive: for NO trio x,y,z, if xRy and yRz, then xRz
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Equivalence relation: a relation that is reflexive, symmetric and transitive
Order is a binary relation which is transitive and in addition either
- (i) reflexive and antisymmetric (weak ordering) or
- (ii) irreflexive and asymmetric (strict ordering)
--------------------------------------------------------------------------------------------------------A BRIEF ASIDE ON LANGUAGE
To use a term, leave it unquoted – this means that you want to go through the term to its
referent; to mention a term, quote it – this means you want to talk about the term itself.
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Seven and ‘7’ have no letters, yet ‘seven’ has five letters and ‘six’ has three
But wait… take ‘seven’. How many letters does it actually have?
(1) Five – 1 ‘s’, 2’e’s, 1 ‘v’, and 1 ‘n’
(2) Four – ‘s’, ‘e’, ‘v’, and ‘n’
In (1), we count the tokens – i.e. the particular instances, regardless of type.
In (2), we count the type – the kinds, regardless of how many instances (as long as >0)
--------------------------------------------------------------------------------------------------------KINDS OF NUMBERS
 Natural: any ordinary whole number, used for counting or ordering. ‘ℕ’ is standardly
used to refer to the set of all natural numbers.
Philosophical Devices
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Infinite Sets & Etc.
Note: some use ‘natural number’ to exclude zero and ‘whole number’ to include it, others
‘whole number’ to excluded zero. We’ll just define the natural numbers as all the nonnegative integers – so zero gets included.
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Integers: any natural numbers (including 0) or a negative of a non-zero natural
numbers; a rational number that can be written without a fractional or decimal
component. ‘ℤ’ is standardly used to refer to the set of all integers
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Rational: any number that can be expressed as the quotient or fraction p/q of two
integers, with the denominator q not equal to zero. ‘ℚ’ is standardly used to refer to
the set of all rational numbers.
Note: since q may be equal to 1, every integer is a rational number (of course, not vice versa).
The decimal expansion of a rational number always either terminates after a finite number of
digits, or begins to repeat the same finite sequence of digits over and over. Any repeating or
terminating decimal represents a rational number. True for base 10, binary, hexadecimal, and
any other integer base! Further, for any two rational numbers, there is another rational
number between them.
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Irrational: any number that cannot be expressed as the quotient or fraction p/q of two
integers, where q is non-zero.
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Real: any number that is rational or irrational. ‘ℝ’ is standardly used to name the set
of all real numbers.
--------------------------------------------------------------------------------------------------------SIZING-UP SETS
First question: When is one set A bigger than another set B?
(1) Whenever set B is a proper subset of A, then A is bigger than B
- {Socrates}  {Socrates, Plato} , so {Socrates, Plato} is bigger than {Plato}
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Cardinality: a measure of the number of elements of the set. For example, set A = {1,
2, 3} contains 3 elements, and therefore has a cardinality of 3.
(2) Whenever any attempt to pair the members of A one-to-one with the members of B
results in some members of A being unpaired.
- Let B= {Socrates}, A={Socrates, Plato}. We can map B’s Socrates to A’s
Socrates, but there is nothing for us to map A’s Plato to. So A must be bigger than
B
- Let A={x| x is a chair in this room}, B= {x| x is a person in this room}. Map each
person in this room to a chair; if there are any chairs left over, then A is bigger
than B
We can associate with each finite set a natural number which represents its cardinality. Sets
with the same cardinality form an equivalence class.
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For all finite sets, these two senses coincide. Not so once we start dealing with infinite set.
In fact, this is the defining characteristic of infinite sets: the members of any infinite set,
but of no finite set, can be paired up one-to-on with the members of some of its proper
Philosophical Devices
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Infinite Sets & Etc.
subsets. (Note: only some, not all! {1, 2, 3}  ℕ, but ℕ cannot be paired one-to-one with
it.)
Second question: Is the set of natural numbers bigger than the set of even numbers?
(1) Yes! Let E={x| x is an even number}. E  ℕ, so there are more naturals than evens
(2) No! Since both are infinite, we’ll never run out of either. This means the evens can be
mapped, one-to-one, with the naturals, and vice versa
Like finite sets, infinite sets can be grouped into equivalence classes, such that all the sets in a
given equivalence class have the same cardinality. We use special symbols which are not
natural numbers to denote the cardinalities of infinite sets; the name for the cardinality of all
sets equivalent to  it is ‘ℵ0’. ℵ0 is not a natural number; it is a cardinal number which is
larger than any natural number.
--------------------------------------------------------------------------------------------------------DENUMERABLE & NON-DENUMERABLE SETS
 Denumerable: an infinite set is denumerable iff it can be mapped, one-to-one with ℕ
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ℤ, the odds, the evens, the squares, the set of all grammatically correct English
sentences, the set of all grammatically correct German sentences…
More interestingly, ℚ: see the grid structure, pg. 24 in Papineau
In fact, there are infinitely many denumerable sets ({x| x is a natural number >2},
{…>3}, {…>4}, etc.)
General strategy: To show that a set is denumerable, find a way to put it into a 1-1
correspondence with the natural numbers. This is called effectively listing the members of the
set, since the natural numbers themselves are thought of as forming an infinite ‘list’.
Now, what about ℝ - is it denumerable?
Premise: the reals between 0 & 1 are denumerable.
If true, then these reals could be mapped, 1-1, with the natural numbers. Granting any such
mapping, we can construct a new number C which has a first digit equal to one more than the
first digit of the first number in our mapping, a second digit equal to one more than the
second digit of the second number in our mapping, a third digit equal to one more than the
third digit of the third number in our mapping…
This new number cannot appear anywhere in our original mapping, since, by definition, for
any number N in the sequence, C will differ from N in the Nth digit – e.g. for the 97th number
in our listing, C will have a 97th digit that is one greater, for the 1037th number, C will have a
1037th digit that is different…
So, by this diagonalization, we know that there must be at least one number that wasn’t
included in our mapping, yet which is a real between 0 &1.
This contradicts our premise that the reals between 0 & 1 are denumerable. Consequently, we
must reject the premise.
Philosophical Devices
Week 2
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Infinite Sets & Etc.
Conclusion: ℝ is non-denumerable. There are more real numbers then natural numbers.
Bigger conclusion: Infinite sets come in different sizes – there are sizes of infinity!
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