Why ordered sequences can be reduced to sets?

Question

I am trying to understand why ordered sequences can be reduced to basic sets.

I understand most of the following proof:

• Sequences can be defined as functions
• Functions are a special case of relations
• and Relations is a special case of sets
• Therefore sequences can be reduced to basic sets (this is what i don't get)

The issue I have is that relations is defined as follows:

A relation Rp between A and B is defined as: Rp ⊆ A × B

Therefore relations makes use of the cartesian product operator, which also makes use of sequences.

Hence relations itself makes use of sequences. Therefore the proof that Sequences can be reduced to sets is flawed.

Please help me understand why sequences can be reduced to sets. Am I correct in my thinking?

Answer 1

Therefore relations makes use of the cartesian product operator, which also makes use of sequences.

What.   Ah... Yes, that is a just wee bit recursive, isn't it now?

The Cartesian product of sets $\rm A, B$ is defined as: $\mathrm A{\times}\mathrm B = \{\langle a,b\rangle \mid a\in\mathrm A \,\wedge\, b\in\mathrm B\}$

That is, it is the set of all ordered pairs whose members are each from the sets under discussion (in the given order).

Now, an ordered pair is a primitive sequence, given that "A sequence is an ordered collection of objects in which repetitions are allowed."

A sequence can be considered a function (mapping index to value).   Such functions can be represented as relations (in this case the set of index, value ordered pairs).   Such a relation is a subset of the Cartesian product of the set of indices and the set of values.   And a Cartesian product is ...

And round and round it goes.

So, really, we can only say: The defined concept of sequences can be reduced to that of basic sets, if we first accept a primitive notion of "ordered pair".

Answer 2

A sequence is a function $f:\mathbb Z^+ \to \mathbf S$.

$f:\mathbb Z^+ \to \mathbf S$ is a special case of a relation $f \subset \mathbb Z^+ \times \mathbf S.$

The cartesian product $\mathbb Z^+ \times \mathbf S$ is the set $\{(x,y): x \in \mathbb Z^+$ and $y \in \mathbf S \}$

I'm not sure what you mean by "basic".

It seems that you are trying to use the category theory definition of direct product, which is a "generalization" of cartesian products but is not the same thing.

Answer 3

There seems to be one step missing:

• ordered pairs (the elements of the cartesian products) can be reduced to sets

Following Kuratowski we can define $$ (x,y):=\{\{x\},\{x,y\}\}$$ which may look somewhat arbitrary, but conveys the essetnial notion of ordered pair: $(x,y)=(u,v)\iff x=u\land y=v$.