Set Theory and ZFC - The Axiom of Replacement - Searcoid, Pages 6-7

[Math Amateur]

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ...

I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ...

I am struggling to attain a full understanding of the Axiom of Replacement which reads as shown below:View attachment 5044
View attachment 5045

I cannot, with any confidence, construct simple examples illustrating the meaning of this axiom ... can someone please help with some simple examples ... note that I am particularly puzzled by the nature, scope and type of conditions \(\displaystyle \phi (x, y )\) that might apply ... what indeed might some \(\displaystyle \phi (x, y )\) look like ... ?

Hope someone can help clarify the meaning of this axiom ...

Peter
NOTE

To enable members of MHB to understand the context and notation of Searcoid's approach to ZFC I am proving the following text relating to some preliminary remarks by the author ...
View attachment 5046
https://www.physicsforums.com/attachments/5047

 

[Evgeny.Makarov]

A programming analogy wold be as follows. Let $A\subset \Bbb N$ and $\phi(n,s)$ mean that $n\in \Bbb N$ and $s$ is a string of $n$ symbols $a$. Then Axiom III applied to $A$ and $\phi$ returns the set of strings of $a$ whose lengths are in $A$.

 

[Math Amateur]

A programming analogy wold be as follows. Let $A\subset \Bbb N$ and $\phi(n,s)$ mean that $n\in \Bbb N$ and $s$ is a string of $n$ symbols $a$. Then Axiom III applied to $A$ and $\phi$ returns the set of strings of $a$ whose lengths are in $A$.

Thanks for the help Evgeny ... since I have been reading a little bit of logic I understand the general sense and direction of your remark ... but ... I am still somewhat puzzled over the Axiom of Replacement ... and also still wondering about the nature/characteristics/scope/possibilities of \(\displaystyle \phi\) ...

... ... so I thought I would construct a couple of simple examples and hope that someone more knowledgeable would critique the examples ... ...

... so in my examples I intend to declare the set \(\displaystyle a\) to be various sets of integers, so a possible \(\displaystyle \phi\) (I think) would be as follows:

\(\displaystyle \phi (x,y)\) is the condition that \(\displaystyle y = x+ 2\) for some \(\displaystyle x \in a \)

... then if ...

\(\displaystyle a\) = set of positive integers = \(\displaystyle \{ 1,2,3, \ ... \ ... \ \}\)

we have, following the Axiom of Replacement ...

\(\displaystyle b = \{ y \ | \ \exists \ x \text{ such that } x \in a \text{ and } \phi (x,y) \text{ holds } \ \}\)

that is

\(\displaystyle b = \{ y \ | \ \exists \ x \text{ such that } x \in a \text{ and } y = x + 2 \ \}\)

so that

\(\displaystyle b = \{ 3, 4, 5, \ ... \ ... \}\)Now, looking at different sets for \(\displaystyle a\) ...

... if \(\displaystyle a\) = set of all even positive integers = \(\displaystyle \{ 2, 4, 6, \ ... \ ... \}\) ... ...

then \(\displaystyle b = \{ 4, 6, 8, \ ... \ ... \}\)

and so on ... that is we can keep altering \(\displaystyle a\) similarly to the above and get the corresponding \(\displaystyle b\) ...Can someone please critique my example above ... either confirming it is correct or pointing out errors and flaws ...


A second question is why is this called the Axiom of Replacement ... is it simply because we are "replacing" a well defined set \(\displaystyle a\) with a new set \(\displaystyle b\) ... is that right? ... but then why is this important ... what is the point?

Another slightly perplexing issue is that my example seems to illustrate both the Axiom of Replacement and the Subset Principle ... is that right? ...

... ... indeed, further, it seems difficult to imagine using the Axiom of Replacement and not coming up with \(\displaystyle b\) as a subset of \(\displaystyle a\) ... is that right? ... does anyone have an example where \(\displaystyle b\) is not a subset, but, indeed, a superset of \(\displaystyle a\)?

... I know that the Subset Principle is derived from the Axiom of Replacement ... but how do they differ?
Another example of the Axiom of Replacement would be to take a as

\(\displaystyle a = \{ 1,2,3, \ ... \ ... \ \}\)

and take \(\displaystyle \phi\) as

\(\displaystyle \phi (x,y)\) is the condition that \(\displaystyle y = \neg (x+ 2)\) for any \(\displaystyle x \in a\)

(that is, \(\displaystyle y\) is the set of integers that is not equal to \(\displaystyle x + 2\) for any \(\displaystyle x \in a\) )

... then we have \(\displaystyle b = \{ 1,2 \}
\)

Is that correct? Can someone please critique my example directly above ... either confirming it is correct or pointing out errors and flaws ...
Hope someone can clarify the issues above and critique my examples ...

Help will be much appreciated ...

Peter
*** EDIT ***

Since the above reply refers to the Axiom of Replacement and the Subset Principle I am providing the text relevant to both directly in this reply, for the convenience of MHB readers ... ... as follows:https://www.physicsforums.com/attachments/5048
https://www.physicsforums.com/attachments/5049View attachment 5050

 

[Math Amateur]

Thanks for the help Evgeny ... since I have been reading a little bit of logic I understand the general sense and direction of your remark ... but ... I am still somewhat puzzled over the Axiom of Replacement ... and also still wondering about the nature/characteristics/scope/possibilities of \(\displaystyle \phi\) ...

... ... so I thought I would construct a couple of simple examples and hope that someone more knowledgeable would critique the examples ... ...

... so in my examples I intend to declare the set \(\displaystyle a\) to be various sets of integers, so a possible \(\displaystyle \phi\) (I think) would be as follows:

\(\displaystyle \phi (x,y)\) is the condition that \(\displaystyle y = x+ 2\) for some \(\displaystyle x \in a \)

... then if ...

\(\displaystyle a\) = set of positive integers = \(\displaystyle \{ 1,2,3, \ ... \ ... \ \}\)

we have, following the Axiom of Replacement ...

\(\displaystyle b = \{ y \ | \ \exists \ x \text{ such that } x \in a \text{ and } \phi (x,y) \text{ holds } \ \}\)

that is

\(\displaystyle b = \{ y \ | \ \exists \ x \text{ such that } x \in a \text{ and } y = x + 2 \ \}\)

so that

\(\displaystyle b = \{ 3, 4, 5, \ ... \ ... \}\)Now, looking at different sets for \(\displaystyle a\) ...

... if \(\displaystyle a\) = set of all even positive integers = \(\displaystyle \{ 2, 4, 6, \ ... \ ... \}\) ... ...

then \(\displaystyle b = \{ 4, 6, 8, \ ... \ ... \}\)

and so on ... that is we can keep altering \(\displaystyle a\) similarly to the above and get the corresponding \(\displaystyle b\) ...Can someone please critique my example above ... either confirming it is correct or pointing out errors and flaws ...


A second question is why is this called the Axiom of Replacement ... is it simply because we are "replacing" a well defined set \(\displaystyle a\) with a new set \(\displaystyle b\) ... is that right? ... but then why is this important ... what is the point?

Another slightly perplexing issue is that my example seems to illustrate both the Axiom of Replacement and the Subset Principle ... is that right? ...

... ... indeed, further, it seems difficult to imagine using the Axiom of Replacement and not coming up with \(\displaystyle b\) as a subset of \(\displaystyle a\) ... is that right? ... does anyone have an example where \(\displaystyle b\) is not a subset, but, indeed, a superset of \(\displaystyle a\)?

... I know that the Subset Principle is derived from the Axiom of Replacement ... but how do they differ?
Another example of the Axiom of Replacement would be to take a as

\(\displaystyle a = \{ 1,2,3, \ ... \ ... \ \}\)

and take \(\displaystyle \phi\) as

\(\displaystyle \phi (x,y)\) is the condition that \(\displaystyle y = \neg (x+ 2)\) for any \(\displaystyle x \in a\)

(that is, \(\displaystyle y\) is the set of integers that is not equal to \(\displaystyle x + 2\) for any \(\displaystyle x \in a\) )

... then we have \(\displaystyle b = \{ 1,2 \}
\)

Is that correct? Can someone please critique my example directly above ... either confirming it is correct or pointing out errors and flaws ...
Hope someone can clarify the issues above and critique my examples ...

Help will be much appreciated ...

Peter
*** EDIT ***

Since the above reply refers to the Axiom of Replacement and the Subset Principle I am providing the text relevant to both directly in this reply, for the convenience of MHB readers ... ... as follows:

I probably should not be replying to my own post ... but I have some thoughts on my concerns when I wrote:

" ... ... ... ... indeed, further, it seems difficult to imagine using the Axiom of Replacement and not coming up with \(\displaystyle b\) as a subset of \(\displaystyle a\) ... is that right? ... does anyone have an example where \(\displaystyle b\) is not a subset, but, indeed, a superset of \(\displaystyle a\)? ... .. "
After some reflection an obvious example became apparent ... and example which if correct shows how it may be possible that in an application of the Axiom of Replacement we can wind up with the set \(\displaystyle b\) not being a subset of the set \(\displaystyle a\) ...

... so ... define \(\displaystyle a = \{ 1,2,3,4 \}\)

and define \(\displaystyle \phi\) as

\(\displaystyle \phi (x,y)\) is the condition that \(\displaystyle y = x+ 2\) for some \(\displaystyle x \in a \)

then ...

\(\displaystyle b = \{ y \ | \ \exists \ x \text{ such that } x \in a \text{ and } \phi (x,y) \text{ holds } \ \}\)

that is

\(\displaystyle b = \{ y \ | \ \exists \ x \text{ such that } x \in a \text{ and } y = x + 2 \ \}\)

so that

\(\displaystyle b = \{ 3, 4, 5, 6 \ ... \ ... \}\)

thus (if the example is correct) we have defined the set b using the Axiom of Replacement and it is not a subset of the set \(\displaystyle a\) ...

Can someone critique my example ... is it correct ...

Mind you, I am a little uneasy about this example ... Evgeny has agreed that we can consider the set \(\displaystyle a\) as a "universe of discourse" ... but my example leads to two objects, namely 5 and 6 that are not in set \(\displaystyle a\), that is not in the universe of discourse ... which does not seem right ...

Can someone please clarify the above ...

Peter

 

[Deveno]

The set $b$ should be $b = \{3,4,5,6\}$, one of the points of the comments Searcoid makes about functional conditions is "the range is no bigger than the domain".

HOWEVER, if $a$ is indeed our "universe of discourse", we are limited to sets that exist in the POWER SET of $a$ (subsets of $a$), and the only $b$ that can thus be specified by your $\phi$ is thus:

$b = \{3,4\}$ (in our "universe" $5$ and $6$ don't "exist").

There is a further problem with your example, one that is probably not immediately evident to you: you use addition in your functional condition. Now, you probably meant for elements of $a$ to be "ordinary integers" and + to mean "ordinary addition", but for addition to even be DEFINED, we require the entirety of the natural numbers (at least). This is because it is defined RECURSIVELY, and thus needs a recursively-defined "ground set" to act on. Indeed, the existence of a such a set cannot be derived from the other ZF axioms in the absence of the axiom of infinity (one can use the axiom of pairing to get arbitrary FINITE set-cardinalities, but one cannot show you can continue this "ad infinitum" and still get a set).

The reason this is a problem is that set theory is often used FOUNDATIONALLY, and one does not use stuff one hasn't "defined" already to define other things. So it the "very beginning" the kinds of functional conditions which one is allowed to use have to not reference "larger sets" than one has already created up to that point.

 

[Math Amateur]

The set $b$ should be $b = \{3,4,5,6\}$, one of the points of the comments Searcoid makes about functional conditions is "the range is no bigger than the domain".

HOWEVER, if $a$ is indeed our "universe of discourse", we are limited to sets that exist in the POWER SET of $a$ (subsets of $a$), and the only $b$ that can thus be specified by your $\phi$ is thus:

$b = \{3,4\}$ (in our "universe" $5$ and $6$ don't "exist").

There is a further problem with your example, one that is probably not immediately evident to you: you use addition in your functional condition. Now, you probably meant for elements of $a$ to be "ordinary integers" and + to mean "ordinary addition", but for addition to even be DEFINED, we require the entirety of the natural numbers (at least). This is because it is defined RECURSIVELY, and thus needs a recursively-defined "ground set" to act on. Indeed, the existence of a such a set cannot be derived from the other ZF axioms in the absence of the axiom of infinity (one can use the axiom of pairing to get arbitrary FINITE set-cardinalities, but one cannot show you can continue this "ad infinitum" and still get a set).

The reason this is a problem is that set theory is often used FOUNDATIONALLY, and one does not use stuff one hasn't "defined" already to define other things. So it the "very beginning" the kinds of functional conditions which one is allowed to use have to not reference "larger sets" than one has already created up to that point.

Hi Deveno ... thanks for the help ...

Just opened your reply ... and saw that you write:

" ... ... The set $b$ should be $b = \{3,4,5,6\}$ ... ..."

Yes, of course ... sorry, meant to write that ... but some very quick ( and careless :( ) cutting and pasting led to the result which you have rightly corrected ... sorry ...

Now focused on the rest of you post ...

Peter

 

[Math Amateur]

The set $b$ should be $b = \{3,4,5,6\}$, one of the points of the comments Searcoid makes about functional conditions is "the range is no bigger than the domain".

HOWEVER, if $a$ is indeed our "universe of discourse", we are limited to sets that exist in the POWER SET of $a$ (subsets of $a$), and the only $b$ that can thus be specified by your $\phi$ is thus:

$b = \{3,4\}$ (in our "universe" $5$ and $6$ don't "exist").

There is a further problem with your example, one that is probably not immediately evident to you: you use addition in your functional condition. Now, you probably meant for elements of $a$ to be "ordinary integers" and + to mean "ordinary addition", but for addition to even be DEFINED, we require the entirety of the natural numbers (at least). This is because it is defined RECURSIVELY, and thus needs a recursively-defined "ground set" to act on. Indeed, the existence of a such a set cannot be derived from the other ZF axioms in the absence of the axiom of infinity (one can use the axiom of pairing to get arbitrary FINITE set-cardinalities, but one cannot show you can continue this "ad infinitum" and still get a set).

The reason this is a problem is that set theory is often used FOUNDATIONALLY, and one does not use stuff one hasn't "defined" already to define other things. So it the "very beginning" the kinds of functional conditions which one is allowed to use have to not reference "larger sets" than one has already created up to that point.

Hi Deveno ... just read your two paragraphs starting with ...

" ... There is a further problem with your example, one that is probably not immediately evident to you ... "

Indeed you were right ... it was not immediately relevant ... BUT ... seems to me you have pointed out something incredibly important that I was missing ... suddenly I am thinking ... there is not much we can use in these logical conditions \(\displaystyle \phi\) ... ...

Thanks again for the help ... your post was indeed quite critical to me gaining an understanding of set theory and ZFC ...

Peter

Peter

 

[Evgeny.Makarov]

a possible \(\displaystyle \phi\) (I think) would be as follows:

\(\displaystyle \phi (x,y)\) is the condition that \(\displaystyle y = x+ 2\) for some \(\displaystyle x \in a \)

Simply "\(\displaystyle \phi (x,y)\) is the condition that \(\displaystyle y = x+ 2\)", period.

Your examples are correct.

A second question is why is this called the Axiom of Replacement

Because the construction in Axiom III replaces every $x\in a$ with the $y$ such that $\phi(x,y)$ holds, if such $y$ exists.

but then why is this important ... what is the point?

Why are functions important, which replace elements of domain with elements of codomain?

Another slightly perplexing issue is that my example seems to illustrate both the Axiom of Replacement and the Subset Principle ... is that right?

No. The axiom of replacement uses a formula $\psi(x,y)$ with two arguments, while the subset principle uses $\phi(x)$ with one argument.

it seems difficult to imagine using the Axiom of Replacement and not coming up with \(\displaystyle b\) as a subset of \(\displaystyle a\)

In my first answer $b$ is a set of strings, which is different from $a$.

does anyone have an example where \(\displaystyle b\) is not a subset, but, indeed, a superset of \(\displaystyle a\)?

Consider $\phi(x,y)$ to be $y=x-1$ (subtraction is on integers) and $a=\{1,2,3,\dots\}$.

take \(\displaystyle \phi\) as

\(\displaystyle \phi (x,y)\) is the condition that \(\displaystyle y = \neg (x+ 2)\) for any \(\displaystyle x \in a\)

The negation $\neg$ can only be applied to propositions, i.e., something that is either true or false. The expression $x+2$ is neither true nor false.

(that is, \(\displaystyle y\) is the set of integers that is not equal to \(\displaystyle x + 2\) for any \(\displaystyle x \in a\) )

Then you should define $\phi(x.y) to be $y=\{z\in\Bbb Z\mid z\ne x+2\}=\Bbb Z\setminus\{x+2\}$.

then we have \(\displaystyle b = \{ 1,2 \}\)

No, each $y$ that makes $\phi(x,y)$ true is a set of integers, so the result of applying the axiom of replacement to it is a family of sets of integers.

Alternatively, you may mean that $\phi(y)$ is $y\in a\land \forall x\in a\;y\ne x+2$. But this property does not depend on $x$.

Evgeny has agreed that we can consider the set a as a "universe of discourse" ... but my example leads to two objects, namely 5 and 6 that are not in set a, that is not in the universe of discourse

That may be why the book authors don't mention the universe of discourse. It's better to think of $a$ as the domain of a function represented by $\phi(x,y)$. And it is not a domain strictly speaking, because the function is partial. The universe of discourse analogy may be applicable to the principle of subset.

This is because it is defined RECURSIVELY, and thus needs a recursively-defined "ground set" to act on. Indeed, the existence of a such a set cannot be derived from the other ZF axioms in the absence of the axiom of infinity

I would not worry about this when one is trying to understand the idea behind the axiom of replacement. After all, we can define addition on a finite set by enumerating its values.

 

[Deveno]

I would not worry about this when one is trying to understand the idea behind the axiom of replacement. After all, we can define addition on a finite set by enumerating its values.

Yes, we can. For example, when we say $y = x + 2$, we could, instead, write:

If $x = 1$, then $y = 3$, if $x = 2$, then $y = 4$, etc. to replace the function:

$y = f(x) = x + 2$ with the explicit correspondence:

$f: 1 \mapsto 3\\2\mapsto 4\\3\mapsto 5\\4\mapsto 6$

which clearly agrees with the function $g: \Bbb N \to \Bbb N$ given by $g(k) = k+2$ when restricted to the set $a = \{1,2,3,4\}$.

However, in *practice*, one wishes to use the axiom of replacement, and in particular the subset principle, when the elements of a set are too numerous to list explicitly. A good example is the following:

Many proofs begin with a statement such as:

"Let $p$ be an odd prime..."

While it is possible to devise a property (predicate) $\phi(x)$ that states:

"$x$ is a odd prime number", and verify that this property holds or not for any *particular* number, I daresay it is well-nigh impossible to define such a property that defines the TOTALITY of all odd primes, without recourse to some larger structure that contains the primes as a subset (typically, this is done in the context of ring theory, but we can use semi-groups, as a more minimal structure).

So we would have to replace all such proofs with "proof schemata". It seems more efficient to use the axiom schemata of ZF theory to reduce the amount of proof-writing we have to do.

That said, you do raise a valid point-sometimes seeing the bigger picture can be confusing to understanding the basic details. I feel, however, that considering these "larger sets" are a bit more illuminating as to the "why do we need these axioms?" sorts of questions. Finite functions, or partially-defined functions, can be enumerated explicitly. The game changes one we pass the stage of being able to list things.

 

[Math Amateur]

Yes, we can. For example, when we say $y = x + 2$, we could, instead, write:

If $x = 1$, then $y = 3$, if $x = 2$, then $y = 4$, etc. to replace the function:

$y = f(x) = x + 2$ with the explicit correspondence:

$f: 1 \mapsto 3\\2\mapsto 4\\3\mapsto 5\\4\mapsto 6$

which clearly agrees with the function $g: \Bbb N \to \Bbb N$ given by $g(k) = k+2$ when restricted to the set $a = \{1,2,3,4\}$.

However, in *practice*, one wishes to use the axiom of replacement, and in particular the subset principle, when the elements of a set are too numerous to list explicitly. A good example is the following:

Many proofs begin with a statement such as:

"Let $p$ be an odd prime..."

While it is possible to devise a property (predicate) $\phi(x)$ that states:

"$x$ is a odd prime number", and verify that this property holds or not for any *particular* number, I daresay it is well-nigh impossible to define such a property that defines the TOTALITY of all odd primes, without recourse to some larger structure that contains the primes as a subset (typically, this is done in the context of ring theory, but we can use semi-groups, as a more minimal structure).

So we would have to replace all such proofs with "proof schemata". It seems more efficient to use the axiom schemata of ZF theory to reduce the amount of proof-writing we have to do.

That said, you do raise a valid point-sometimes seeing the bigger picture can be confusing to understanding the basic details. I feel, however, that considering these "larger sets" are a bit more illuminating as to the "why do we need these axioms?" sorts of questions. Finite functions, or partially-defined functions, can be enumerated explicitly. The game changes one we pass the stage of being able to list things.

Hi Deveno, Evgeny ... thanks for all your help/guidance and thoughts on ZFC ...

Still reflecting on all that you have said ...

Your helpful and thoughtful posts make the MHB a real learning environment ...

Thanks again.

Peter*** EDIT ***

Deveno mentioned ur-elements in one of his posts ... maybe we need some theory that includes ur-elements ... but how this would work, I have no idea ... but presumably it would be no more difficult than trying to imagine a universe where everything is a set, including the elements of sets ...kind of like "turtles all the way down" ... ...

Peter

 

[Math Amateur]

Hi Deveno, Evgeny ... thanks for all your help/guidance and thoughts on ZFC ...

Still reflecting on all that you have said ...

Your helpful and thoughtful posts make the MHB a real learning environment ...

Thanks again.

Peter

Deveno, Evgeny ... read all your comments ... thanks again ...

BUT ... another thing bothers me about my examples ... Searcoid states that under ZFC all objects are sets ... but ? ... in my examples I am using integers such as 1, 2, 3 etc ... but these are not sets ... ? ... indeed it seems difficult to frame examples where everything is a set ...

Can you comment and perhaps clarify this area of concern ...?

Peter

 

[Deveno]

In a "standard" (read: von Neumann) construction of the natural numbers, the set $\Bbb N$ is defined like so:

Let $\Phi(x)$ be the statement:

$(\emptyset \in x \wedge \forall y(y \in x \implies y \cup \{y\} \in x))$

Loosely interpreted, this says: "$x$ is an inductive set."

The axiom of infinity asserts that there is at least one set $I$ such that $\Phi(I)$ is true.

We now define:

$\Bbb N = \{x \in I: \forall J(\Phi(J) \implies x \in J)\}$

In other words, we are essentially taking the intersection of every inductive set (the "$J$" 's) to "throw out" any non-natural number elements that might exist in our given inductive set $I$. What allows us to do this is the axiom schema of specification (we are specifying a sub-class of a set).

Now, by construction, we have:

$\emptyset \in \Bbb N$, which we henceforth refer to by the *name* $0$.

By construction, we also have $\emptyset \cup \{\emptyset\} = \{\emptyset\} \in \Bbb N$. We henceforth refer to this by the *name* $1$.

That is, $1$ is the set: $\{0\}$.

Similarly, we give the *name* $2$ to the set $1 \cup \{1\} = \{0,1\}$.

The "general formula" is- to the set:

$n-1 \cup \{n-1\} = \{0,1,2,\dots,n-1\}$, we give the *name* $n$.

Thus every natural number can be seen to be a set which we build up "from scratch".

Note we can define an ORDER on the naturals by:

$n \leq m \iff n \subseteq m$.

This order makes itself felt through many "extensions" of the natural numbers.

I should note in passing that there are, in fact, other ways to create inductive sets, one of the most well-known is the following:

$\emptyset = 0$
$\{0\} = 1$
$\{1\} = 2$, etc. (this construction is due to Zermelo).

This fact is often used to undercut the position of Platonic math philosophers, for if numbers are "truly existent entities" than both of these definitions cannot simultaneously be correct (there are other subtleties involved, but that's the gist of the argument).

From these humble beginnings, it's then not hard to show that the integers, the rationals, the reals, the complexes, etc. are all well-defined sets, and as such we may specify any singleton subset as a set in its own right. This gives quite a few sets, but it doesn't end there-we can create vector spaces, groups, rings, fields, algebras of many shapes and sizes (often using sets of functions, or relations). Topology focuses on certain subsets of a power set for a given set (i.e. the "space"), indeed just the set of all subsets of real numbers is so *rich* that we can impose many different kinds of structures upon it (cantor sets, open sets, sequence sets, intervals, borel sets...just to name a few).

In much of mathematics, as soon as "enough sets are developed" to allow us to "forget their origins" and just apply the set-theoretic analogues of logic to them:

$\vee \leftrightarrow \cup$
$\wedge \leftrightarrow \cap$
$\implies \leftrightarrow\ \subseteq$
$\neg \leftrightarrow (-)^c$

often the underlying formalism is dispensed with entirely (or perhaps even ignored at the outset).

 

[Math Amateur]

In a "standard" (read: von Neumann) construction of the natural numbers, the set $\Bbb N$ is defined like so:

Let $\Phi(x)$ be the statement:

$(\emptyset \in x \wedge \forall y(y \in x \implies y \cup \{y\} \in x))$

Loosely interpreted, this says: "$x$ is an inductive set."

The axiom of infinity asserts that there is at least one set $I$ such that $\Phi(I)$ is true.

We now define:

$\Bbb N = \{x \in I: \forall J(\Phi(J) \implies x \in J)\}$

In other words, we are essentially taking the intersection of every inductive set (the "$J$" 's) to "throw out" any non-natural number elements that might exist in our given inductive set $I$. What allows us to do this is the axiom schema of specification (we are specifying a sub-class of a set).

Now, by construction, we have:

$\emptyset \in \Bbb N$, which we henceforth refer to by the *name* $0$.

By construction, we also have $\emptyset \cup \{\emptyset\} = \{\emptyset\} \in \Bbb N$. We henceforth refer to this by the *name* $1$.

That is, $1$ is the set: $\{0\}$.

Similarly, we give the *name* $2$ to the set $1 \cup \{1\} = \{0,1\}$.

The "general formula" is- to the set:

$n-1 \cup \{n-1\} = \{0,1,2,\dots,n-1\}$, we give the *name* $n$.

Thus every natural number can be seen to be a set which we build up "from scratch".

Note we can define an ORDER on the naturals by:

$n \leq m \iff n \subseteq m$.

This order makes itself felt through many "extensions" of the natural numbers.

I should note in passing that there are, in fact, other ways to create inductive sets, one of the most well-known is the following:

$\emptyset = 0$
$\{0\} = 1$
$\{1\} = 2$, etc. (this construction is due to Zermelo).

This fact is often used to undercut the position of Platonic math philosophers, for if numbers are "truly existent entities" than both of these definitions cannot simultaneously be correct (there are other subtleties involved, but that's the gist of the argument).

From these humble beginnings, it's then not hard to show that the integers, the rationals, the reals, the complexes, etc. are all well-defined sets, and as such we may specify any singleton subset as a set in its own right. This gives quite a few sets, but it doesn't end there-we can create vector spaces, groups, rings, fields, algebras of many shapes and sizes (often using sets of functions, or relations). Topology focuses on certain subsets of a power set for a given set (i.e. the "space"), indeed just the set of all subsets of real numbers is so *rich* that we can impose many different kinds of structures upon it (cantor sets, open sets, sequence sets, intervals, borel sets...just to name a few).

In much of mathematics, as soon as "enough sets are developed" to allow us to "forget their origins" and just apply the set-theoretic analogues of logic to them:

$\vee \leftrightarrow \cup$
$\wedge \leftrightarrow \cap$
$\implies \leftrightarrow\ \subseteq$
$\neg \leftrightarrow (-)^c$

often the underlying formalism is dispensed with entirely (or perhaps even ignored at the outset).

Thanks Deveno ... still reflecting on the above post ...

Taking a while to get my mind around some of the notions you outline and explain ... ... ... ...

Indeed it looks as if you may have explained my questions on the Axiom of Infinity but I am still trying to get my mind around (and understand the mechanics of the expressions (and associated statement)

" ... ... Let $\Phi(x)$ be the statement:

$(\emptyset \in x \wedge \forall y(y \in x \implies y \cup \{y\} \in x))$

Loosely interpreted, this says: "$x$ is an inductive set. ... ... "


and " ... ... $\Bbb N = \{x \in I: \forall J(\Phi(J) \implies x \in J)\}$ ... ... "

Peter

 

[Math Amateur]

In a "standard" (read: von Neumann) construction of the natural numbers, the set $\Bbb N$ is defined like so:

Let $\Phi(x)$ be the statement:

$(\emptyset \in x \wedge \forall y(y \in x \implies y \cup \{y\} \in x))$

Loosely interpreted, this says: "$x$ is an inductive set."

The axiom of infinity asserts that there is at least one set $I$ such that $\Phi(I)$ is true.

We now define:

$\Bbb N = \{x \in I: \forall J(\Phi(J) \implies x \in J)\}$

In other words, we are essentially taking the intersection of every inductive set (the "$J$" 's) to "throw out" any non-natural number elements that might exist in our given inductive set $I$. What allows us to do this is the axiom schema of specification (we are specifying a sub-class of a set).

Now, by construction, we have:

$\emptyset \in \Bbb N$, which we henceforth refer to by the *name* $0$.

By construction, we also have $\emptyset \cup \{\emptyset\} = \{\emptyset\} \in \Bbb N$. We henceforth refer to this by the *name* $1$.

That is, $1$ is the set: $\{0\}$.

Similarly, we give the *name* $2$ to the set $1 \cup \{1\} = \{0,1\}$.

The "general formula" is- to the set:

$n-1 \cup \{n-1\} = \{0,1,2,\dots,n-1\}$, we give the *name* $n$.

Thus every natural number can be seen to be a set which we build up "from scratch".

Note we can define an ORDER on the naturals by:

$n \leq m \iff n \subseteq m$.

This order makes itself felt through many "extensions" of the natural numbers.

I should note in passing that there are, in fact, other ways to create inductive sets, one of the most well-known is the following:

$\emptyset = 0$
$\{0\} = 1$
$\{1\} = 2$, etc. (this construction is due to Zermelo).

This fact is often used to undercut the position of Platonic math philosophers, for if numbers are "truly existent entities" than both of these definitions cannot simultaneously be correct (there are other subtleties involved, but that's the gist of the argument).

From these humble beginnings, it's then not hard to show that the integers, the rationals, the reals, the complexes, etc. are all well-defined sets, and as such we may specify any singleton subset as a set in its own right. This gives quite a few sets, but it doesn't end there-we can create vector spaces, groups, rings, fields, algebras of many shapes and sizes (often using sets of functions, or relations). Topology focuses on certain subsets of a power set for a given set (i.e. the "space"), indeed just the set of all subsets of real numbers is so *rich* that we can impose many different kinds of structures upon it (cantor sets, open sets, sequence sets, intervals, borel sets...just to name a few).

In much of mathematics, as soon as "enough sets are developed" to allow us to "forget their origins" and just apply the set-theoretic analogues of logic to them:

$\vee \leftrightarrow \cup$
$\wedge \leftrightarrow \cap$
$\implies \leftrightarrow\ \subseteq$
$\neg \leftrightarrow (-)^c$

often the underlying formalism is dispensed with entirely (or perhaps even ignored at the outset).

Hi Deveno ... just reflecting on your post above ... and ... need some further help and clarification ...

You write:

" ... ... Let $\Phi(x)$ be the statement:

$(\emptyset \in x \wedge \forall y(y \in x \implies y \cup \{y\} \in x))$ ... ... "

I am trying to ensure I make sense of the condition \Phi ...

In \(\displaystyle \Phi(x)\) we definitely have that \(\displaystyle \phi \in x\) ... and then we turn to the statement after the 'and' and using this we have ...

\(\displaystyle \phi \in x \Longrightarrow \phi \cup \{ \phi \} \in x \)

but then we apply the second condition after the 'and' again and we have

\(\displaystyle \phi \cup \{ \phi \} \in x \Longrightarrow \phi \cup \{ \phi \} \cup \{ \phi \cup \{ \phi \} \} \) \(\displaystyle \in x \) ... ...

and so on ...

Is that right ...?Another ... probably dumb question ... why is \(\displaystyle \phi \cup \{ \phi \} = \{ \phi \}\) ... ?

I know it sounds extremely plausible ... but how do we rigorously prove this ... ?

Hope you can help ...

Peter

 

[Deveno]

For any set $a$, we have $a \cup \emptyset = a$.

The empty set has no members, so the set:

all the elements of $a$, and all the (and there are none!) elements of the empty set

is the same as the set:

all the elements of $a$.

Put another way, it is obvious that:

$a \subseteq a \cup \emptyset$ (this is true of unions in general).

Now suppose $x \in (a \cup \emptyset)$. If $x \in a$, then certainly $x \in a$.

Else, $x \in \emptyset$. Now there are two ways to proceed from here:

1. Observe that $x \in \emptyset$ is never true, since $\emptyset$ is empty.

2. Observe that $\emptyset \subseteq a$, for *any* set $a$. This leads us to:

$a \cup \emptyset \subseteq a \cup a = a$.

(If something is in $a$, or...it's in $a$, I think it safe to conclude it's in $a$)

Either way we have $(a \subseteq a \cup \emptyset) \wedge (a \cup \emptyset \subseteq a)$, so since these two sets have precisely the same elements, they are equal (axiom of extensionality).