Version 5 after 4 very rough drafts
Aim
We use a Peano-like approach to build "proto-integers" from
"proto-sets." These proto-integers are then used to justify numerical
quantifiers that can then be applied to sets forthwith,
alongside the standard
∀ and ∃ (and perhaps ∃!) and accepting the
"not" sign, ~ .
As we shall see, however, it is not required that we call our objects "sets" -- although psychologically it is hard to distinguish them from some sort of set or other, as the next paragraph indicates. Though it is a useful distinction elsewhere, no attempt is made to distinguish the word "set" from "class."
Once we axiomatize our system, we find implied a priori proto-sets. These are well justified by Quine's argument concerning how children learn abstraction and communication. That is, a word represents an idea that does not precisely match every imagined instance. So it becomes necessary to say that a word represents an idea associated with other ideas or images. These secondary ideas are known as properties or attributes. So a word represents an abstracted idea, shorn of the potential distinctive properties.
From this basic law of thought, the idea of collection, class or set must follow. So we are entitled to accept such primitive sets as self-evident. Beyond this we may not go without formulating a proper set theory with an associated system of logic. But what we may do is apply the numerical quantifiers to these primitive sets right away. We don't need to establish the foundations of arithmetic in terms of a proper class theory or to define numbers with formal sets, as in the von Neumann derivation of integers.
Our method does in fact derive the basic arithmetical operations, but this is a frill that does not affect the basic aim of our approach. We indulge in anachronisms when the method is applied to "advanced" systems for which we have not troubled to lay the groundwork.
So once we have these integer quantifiers, we may then go on to establish some formal class theory or other, such as ZFC or NBG. If we like, we can at some point dispense with the proto-integers and accept, say, the von Neumann set theoretic definition of integer.
There is nothing terribly novel in this method. The point is to show that we can accompany basic set theory with exact integer quantifiers right away.
Method
Some Euclidean axioms that we appropriate
We define "0" magnitude as the distance covered by the intersection of two lines.
Now, for graphic purposes, we shall imagine an S perfectly inscribed within a square. The "S" is for our convenience; it is the square upon which we rely.
Now as we consider a strip of squares (say beginning on the right with the eye going leftward), we observe that there is a vertical line at the beginning on the right. Beyond that there is no square. We shall symbolize that condition with a "0" .
At this point we interject that the term "adjacent square" means that squares have a common side and that we are pointing to, or designating, a specific square.
The word "consecutive" implies that there is some strip of squares such that if we examine any specific square we find that it is always adjacent to some other(s). To be more precise, we use the concepts of "left" and "right," which, like "top" and "bottom," are not defined. If at the leftmost or rightmost side of a strip of squares (or "top or bottom"), we find there is no adjacent square, we may use that extreme square as a "beginning." We then sheer off only that square.
We then repeat the process. There is now a new "beginning" square that meets the original conditions. This is then shorn off. This algorithm may be performed repeatedly. This process establishes the notions of "consecutive" and "consecutive order." If there is no "halt" order implied, then the process is open-ended. We cannot say that an infinity is really implied, as we have not got to more advanced class theory (which we are not going to do).
We can say that a halt order is implied whenever we have decided to name a strip, which becomes obvious from what follows.
So then, all this permits us to use the unoriginal symbolism
S0
Under Peano's rule, 0 has no predecessor and every S has a predecessor.
From this we obtain 0 --> no square to the right.
S0 is the successor of 0 and is named "1."
SS0 is called the successor of "1" and is named "2."
From here we may justify "any" constructible integer without resort to mathematical induction, an axiom of infinity and an infinite axiom scheme. We do not take the word "any" as it is used with the "all" quantifier. Rather, what we mean is that if a number is constructible by the open-ended successor algorithm, it can also be used for counting purposes.
Now we derive the arithmetical operations.
Addition
Example
"1 + 2 = 3" is justified thus:
We write S0 + SS0, retaining the plus sign as convenient.
This tells us to eliminate or ignore the interior '0' and slide the left-hand S (or S's, as the case may be) to the right, thus giving the figure
SSS0.
We can, if desired, be fussy and not talk about sliding S's but about requiring that the strip S0 must extend leftward from the leftward vertical side of strip SS0 on ground that 0 implies no distance between the two strips.
Subtraction
Example
5 - 2 = 3
Subtraction is handled by first forming a third horizontal parallel line that is also a unit distance apart from the nearest other parallel. Thus we have two rows of squares that can be designated by S place-holders.
We write
where we have designated with 0's those squares on the second strip that are to the left of the bottom successor strip and directly under squares of the top successor strip. Hence, we require that only vertical strips with no 0's be erased and collapsed (again, we could be more finicky in defining "erase and collapse" but won't be bothered).
The result is
SSS[SS]0 = SSS0
Note that we may reverse the procedure to obtain negative numbers.
A negative number is defined for K - J with K < J. The less-than relation is determined if we have two strips, as above, in which one strip contains leftward 0's. The strip with the leftward 0's is "less than" the strip without leftward 0's.
So then,
1 - 2 = -1
results from
Similarly we cross out the vertical S's, preserving the top strip, which gives
0S0
Though that last expression is OK, for symmetry, we should drop the right-hand 0. In that case 0S is -1, 0SS is -2, and so on.
We require (this must be an axiom) that
Axiom: 0S + S0 = 00 = 0
In which case, we may reduce matching opposed numbers K + -K to the form
0S
S0
which is 0.
For example, 0SS + SS0 gives
and by erasing the columns of S's (no 0's), we obtain the 0 identity.
Multiplication
Example
3 x 2 = 6
We decide that we will associate the left of the multiplication sign with horizontal rows and the right with vertical columns.
Thus,
We match each horizontal "2" with a strip under it, until we have reached the vertical number "3." As a nicety, I have required a bottom row of 0's, to assure that the columnular number is defined. We then slide each row onto the top framework of squares, thus:
SS0SS0SS0
However, interior 0's imply that there is no distance between sub-strips. Hence we erase them and of course get
SSSSSS0
which we have decided to name "6."
Division
Exact division
i) If two strips are identical we say that only one name is to be assigned -- say, "K." That is, they both take the same name. Thus if two strips completely match (no difference in magnitude), then the number K is said to divide by K.
ii) Let a shorter strip be placed under another strip.
The shorter strip is said to divide exactly the longer if the shorter strip is replicated and placed leftward under the longer strip and, after erasure of interior 0's, the two row strips are identical.
But before we may do that, we must ascertain what number the exact division yields. In other words, we have proved that 2 divides 4 exactly. But we have not shown that 4/2 = 2. This requires another step, which harks back to the multiplication procedure.
Each sub-strip in row 2 must be placed in a new row, using the former row 2 as the present top row, and, for clarity, we add a row of 0's. From the above example:
We may now read down the left-hand column to obtain the desired divisor.
SSSSSSSSS0
divided by the strip SSS0 yields
By this, we have proved that the number known as 9 is exactly divided by the number called 3 into 3 strips, all with the name 3.
Rationals
Rationals are defined by putting one successor number atop another and calling it a ratio.
We do not permit (axiom) division by 0 or, that is, for a ratio's denominator to have no predecessor.
is 2/3 and likewise,
is 3/2.
and
yields - 1/2
while
yields + 1
and
yields + 1/2
We do not enter into the subject of equivalence classes, which at this point would be a highly anachronistic topic.
So then we can say, for example, (2x ∈ X)(x,a), which reads there are at least 2 x in X which have the property a. Of course, that does not mean we are not obliged to build up the sets and propositions in some coherent fashion.
By establishing proto-integers through the use of some routine axioms, we are able to give exact quantifiers for any sets we intend to build. As a bonus, we have established the basic arithmetic operations without resort to formal set theory.
The definition of successor/predecessor relation is easily derived from the discussion above of "consecutive" and "next."
For our purposes, a proto-set, or "set," is a successor number the elements of which are predecessors (so this is similar to the Von Neumann method in which a successor is defined "x U {x}." Our method, I would say, is a tad more primitive.
Our "elements" may be visualized by placing each immediate predecessor on an adjacent horizontal strip, as shown:
Or we may have an equivalent graph
which is handy because we now have a matrix with its row and column vectors -- though of course we are not anywhere near that level of abstraction in our specific business.
So, if we like, we may denote each strip with the name "set." The bottom strip we call the "0 set" which means that it is the class with no predecessor. That it is equivalent to the empty set of standard class theories is evident because it implies no predecessor, which is to say there can be no "element" beneath it. Also, note that Russell's paradox does not arise in this primitive system, because a strip number's "element" (we are free to avoid that loaded word if we choose) strips are always below it.
Now note that the top number has an S on the extreme left -- rather than a 0 -- such that S3 ⇒ S2 ⇒ S1 ⇒ 0 (where the sub numbers are only for our immediate convenience and have no intrinsic meaning; we could as well use prime marks or arbitrary names, as in Sdog ⇒ Sstarship.
In any case we may, only if we so desire, name the entire graphic above as a "set" or "proto-set." Similarly, we may so name each sub-graphic that occurs when we erase a top strip. Obviously, this parallels the usual set succession rules.
Though our naming these graphics "sets" is somewhat user-friendly, it is plainly unnecessary. We could as well name them with some random string of characters, say "xxjtdbn." The entire graph has the general name "xxjtdbn." Under it is another xxjtdbn, which differs from the other and so must take another name as a mark of distinction. In fact, every permissible graph must take some distinct name.
So for the entire graph we have "xxjtdbn." For the "next" sub-graph, we have "agbfsaf." For the one below that, we have "dtdmitg." And for the "0" strip we have "zbhikeb."
We are expected to know that each name applies to a specific strip and thus is either a successor or a predecessor or both. So then, we don't really need to employ the abstract concept of "set" (though we are employing abstract Euclidean axioms and a couple of other axioms).
Now if we write, for example,
(2x ∈ X)P
we seem to be saying that the more "advanced" set definitions are in force for "X" and so "2x ∈ X" is not a legitimate quantifier of the assertion (=proposition) P.
It is true that we are not done with our quantifier design. We are saying that "2" is a name given our graphic that is also known as "agbfsaf." We accept that there may be objects of some sort that go by the generic name "x." We must be able to establish a 1-1 correspondence between our agbfsaf graphic and any x's.
That is, we must be able to draw a single line (at least notionally) between every strip of agbfsaf, except the 0 strip, and one and only one of the x's. If that 1-1 correspondence is exact -- no unconnected ends -- then we may write (2!x ∈ X)P, which tells us that there are exactly 2 x's that apply to P "truthfully."
We have implied in our notation that there is a class X of which the x's are members. This isn't quite necessary. We may just say that X is a name for various x's. We might even say that we have a simple pairing system (aka "binary relation," though we must beware this terminology as probably anachronistic) such that xo,X, meaning that x may vary but that X may not.
Now suppose we wish to talk about equality, as in P means "x = x."
We may write (∀ x)(x = x), (∃ x)(x = x) and ~(∀ x)(x = x) in standard form.
Now if we mean by x one of the graphic "numbers," each graph shows a 1-1 correspondence with a copy of itself and so we can establish the first two statements as true and the last as false. If we mean that x represents arbitrary ZFC or NBG class theoretic numbers, we can still use the correspondence test (though we need not deal with indefinite unbounded algorithms -- loosely dubbed "infinity").
Further, we may use the correspondence test for, say, the graphic named "2." If we find that we can draw single lines connecting strips of "2" with objects known as x's among NBG or ZFC "numbers," then we can say 2x(x = x). That means that graphic "2" is 1-1 with some part of ZFC or NBG. Or, we would normally say, "there are at least 2 x's in NBG or ZFC that are equal to themselves (where "self" = "duplicate")," as opposed to 2!x(x = x), which would normally be verbalized as "there are exactly 2 x's in NBG or ZFC that are equal to themselves (or their duplicates). The last statement can only be true if we specify what x's we are talking about.
I concede that in these last few paragraphs I mix apples and oranges. Why would we need the graphic "numbers" if we already have ZFC or NBG? But, we do have proof of principle -- that much can be done with these successor graphics, or what some might term pseudo-sets.
1. In his book Logic for Mathematicians (Chelsea Publishing 1978, McGraw Hill 1953), J. Barkley Rosser cautions against the word any. "Sometimes any means each and sometimes it means some. Thus, sometimes "for any x..." means (x) and sometimes "for any x..." means (Ex)."
¶ [Note: (x) is an old-fashioned way to denote ∀x and (Ex) is Rosser's way of denoting ∃x.]
¶ After giving an ambiguous example, Rosser says, "If one wishes to be sure that one will be understood, one should never use any in a place where either each or some can be used. For instance, "I will not violate any law." The statement "I will not violate each law" has quite a different meaning, and the statement, "I will not violate some law" might be interpreted to mean that there is a particular law which I am determined not to violate.
¶ "Nonetheless," says Rosser, "many writers use any in places where each or some would be preferable."
As we shall see, however, it is not required that we call our objects "sets" -- although psychologically it is hard to distinguish them from some sort of set or other, as the next paragraph indicates. Though it is a useful distinction elsewhere, no attempt is made to distinguish the word "set" from "class."
Once we axiomatize our system, we find implied a priori proto-sets. These are well justified by Quine's argument concerning how children learn abstraction and communication. That is, a word represents an idea that does not precisely match every imagined instance. So it becomes necessary to say that a word represents an idea associated with other ideas or images. These secondary ideas are known as properties or attributes. So a word represents an abstracted idea, shorn of the potential distinctive properties.
From this basic law of thought, the idea of collection, class or set must follow. So we are entitled to accept such primitive sets as self-evident. Beyond this we may not go without formulating a proper set theory with an associated system of logic. But what we may do is apply the numerical quantifiers to these primitive sets right away. We don't need to establish the foundations of arithmetic in terms of a proper class theory or to define numbers with formal sets, as in the von Neumann derivation of integers.
Our method does in fact derive the basic arithmetical operations, but this is a frill that does not affect the basic aim of our approach. We indulge in anachronisms when the method is applied to "advanced" systems for which we have not troubled to lay the groundwork.
So once we have these integer quantifiers, we may then go on to establish some formal class theory or other, such as ZFC or NBG. If we like, we can at some point dispense with the proto-integers and accept, say, the von Neumann set theoretic definition of integer.
There is nothing terribly novel in this method. The point is to show that we can accompany basic set theory with exact integer quantifiers right away.
Method
Some Euclidean axioms that we appropriate
1. A line on the plane may be intersected by two other lines A and B such that the distance from A to B is definite (measurable in principle with some yardstick) which we shall call magnitude AB.From these, we are able to construct and imply two parallel lines A and B, each of which is intersected by lines of a "unit" length apart and we can arrange that the perpendicular distance from A to B is unity. In other words, we have a finite strip of squares all lined up. We have given no injunction against adding more squares along the horizontals.
2. Magnitude, or distance, AB can be exactly duplicated with another intersecting line C such that AB and CA have no distance between their nearer interior end points.
3. Some line A may be intersected by a line B such that A and B are perpendicular.
We define "0" magnitude as the distance covered by the intersection of two lines.
Now, for graphic purposes, we shall imagine an S perfectly inscribed within a square. The "S" is for our convenience; it is the square upon which we rely.
Now as we consider a strip of squares (say beginning on the right with the eye going leftward), we observe that there is a vertical line at the beginning on the right. Beyond that there is no square. We shall symbolize that condition with a "0" .
At this point we interject that the term "adjacent square" means that squares have a common side and that we are pointing to, or designating, a specific square.
The word "consecutive" implies that there is some strip of squares such that if we examine any specific square we find that it is always adjacent to some other(s). To be more precise, we use the concepts of "left" and "right," which, like "top" and "bottom," are not defined. If at the leftmost or rightmost side of a strip of squares (or "top or bottom"), we find there is no adjacent square, we may use that extreme square as a "beginning." We then sheer off only that square.
We then repeat the process. There is now a new "beginning" square that meets the original conditions. This is then shorn off. This algorithm may be performed repeatedly. This process establishes the notions of "consecutive" and "consecutive order." If there is no "halt" order implied, then the process is open-ended. We cannot say that an infinity is really implied, as we have not got to more advanced class theory (which we are not going to do).
We can say that a halt order is implied whenever we have decided to name a strip, which becomes obvious from what follows.
So then, all this permits us to use the unoriginal symbolism
S0
Under Peano's rule, 0 has no predecessor and every S has a predecessor.
From this we obtain 0 --> no square to the right.
S0 is the successor of 0 and is named "1."
SS0 is called the successor of "1" and is named "2."
From here we may justify "any" constructible integer without resort to mathematical induction, an axiom of infinity and an infinite axiom scheme. We do not take the word "any" as it is used with the "all" quantifier. Rather, what we mean is that if a number is constructible by the open-ended successor algorithm, it can also be used for counting purposes.
Now we derive the arithmetical operations.
Addition
Example
"1 + 2 = 3" is justified thus:
We write S0 + SS0, retaining the plus sign as convenient.
This tells us to eliminate or ignore the interior '0' and slide the left-hand S (or S's, as the case may be) to the right, thus giving the figure
SSS0.
We can, if desired, be fussy and not talk about sliding S's but about requiring that the strip S0 must extend leftward from the leftward vertical side of strip SS0 on ground that 0 implies no distance between the two strips.
"Two" is not defined here as a number, but as a necessary essential idea that we use to mean a specified object of attention and an other specified object of attention. I grant that the article "an" already implies "oneness" and the word "other" already implies "twoness." Yet these are "proto-ideas" and not necessarily numbers. BUT, since we have actually defined numbers by our successor algorithm, we are now free to apply them to our arithmetic operations.
Subtraction
Example
5 - 2 = 3
Subtraction is handled by first forming a third horizontal parallel line that is also a unit distance apart from the nearest other parallel. Thus we have two rows of squares that can be designated by S place-holders.
We write
SSSSS0 000SS0
where we have designated with 0's those squares on the second strip that are to the left of the bottom successor strip and directly under squares of the top successor strip. Hence, we require that only vertical strips with no 0's be erased and collapsed (again, we could be more finicky in defining "erase and collapse" but won't be bothered).
The result is
SSS[SS]0 = SSS0
Note that we may reverse the procedure to obtain negative numbers.
A negative number is defined for K - J with K < J. The less-than relation is determined if we have two strips, as above, in which one strip contains leftward 0's. The strip with the leftward 0's is "less than" the strip without leftward 0's.
So then,
1 - 2 = -1
results from
0S0 SS0
Similarly we cross out the vertical S's, preserving the top strip, which gives
0S0
Though that last expression is OK, for symmetry, we should drop the right-hand 0. In that case 0S is -1, 0SS is -2, and so on.
We require (this must be an axiom) that
Axiom: 0S + S0 = 00 = 0
In which case, we may reduce matching opposed numbers K + -K to the form
0S
S0
which is 0.
For example, 0SS + SS0 gives
0SSS
SSS0
and by erasing the columns of S's (no 0's), we obtain the 0 identity.
Multiplication
Example
3 x 2 = 6
We decide that we will associate the left of the multiplication sign with horizontal rows and the right with vertical columns.
Thus,
SS0 SS0 SS0 000
We match each horizontal "2" with a strip under it, until we have reached the vertical number "3." As a nicety, I have required a bottom row of 0's, to assure that the columnular number is defined. We then slide each row onto the top framework of squares, thus:
SS0SS0SS0
However, interior 0's imply that there is no distance between sub-strips. Hence we erase them and of course get
SSSSSS0
which we have decided to name "6."
Division
Exact division
i) If two strips are identical we say that only one name is to be assigned -- say, "K." That is, they both take the same name. Thus if two strips completely match (no difference in magnitude), then the number K is said to divide by K.
ii) Let a shorter strip be placed under another strip.
SSSS0 00SS0
The shorter strip is said to divide exactly the longer if the shorter strip is replicated and placed leftward under the longer strip and, after erasure of interior 0's, the two row strips are identical.
SSSS0 SSSS0
But before we may do that, we must ascertain what number the exact division yields. In other words, we have proved that 2 divides 4 exactly. But we have not shown that 4/2 = 2. This requires another step, which harks back to the multiplication procedure.
Each sub-strip in row 2 must be placed in a new row, using the former row 2 as the present top row, and, for clarity, we add a row of 0's. From the above example:
SS0 SS0 000
We may now read down the left-hand column to obtain the desired divisor.
SSSSSSSSS0
divided by the strip SSS0 yields
SSS0 SSS0 SSS0 0000
By this, we have proved that the number known as 9 is exactly divided by the number called 3 into 3 strips, all with the name 3.
Rationals
Rationals are defined by putting one successor number atop another and calling it a ratio.
We do not permit (axiom) division by 0 or, that is, for a ratio's denominator to have no predecessor.
0SS0 SSS0
is 2/3 and likewise,
SSS0 0SS0
is 3/2.
and
0S0 SS0
yields - 1/2
while
0S0 0S0
yields + 1
and
00S0 0SS0
yields + 1/2
We do not enter into the subject of equivalence classes, which at this point would be a highly anachronistic topic.
So then we can say, for example, (2x ∈ X)(x,a), which reads there are at least 2 x in X which have the property a. Of course, that does not mean we are not obliged to build up the sets and propositions in some coherent fashion.
By establishing proto-integers through the use of some routine axioms, we are able to give exact quantifiers for any sets we intend to build. As a bonus, we have established the basic arithmetic operations without resort to formal set theory.
The definition of successor/predecessor relation is easily derived from the discussion above of "consecutive" and "next."
For our purposes, a proto-set, or "set," is a successor number the elements of which are predecessors (so this is similar to the Von Neumann method in which a successor is defined "x U {x}." Our method, I would say, is a tad more primitive.
Our "elements" may be visualized by placing each immediate predecessor on an adjacent horizontal strip, as shown:
S S S 0
S S 0 S 0 0
Or we may have an equivalent graph
S S S 0 0 S S 0 0 0 S 0 0 0 0 0
which is handy because we now have a matrix with its row and column vectors -- though of course we are not anywhere near that level of abstraction in our specific business.
So, if we like, we may denote each strip with the name "set." The bottom strip we call the "0 set" which means that it is the class with no predecessor. That it is equivalent to the empty set of standard class theories is evident because it implies no predecessor, which is to say there can be no "element" beneath it. Also, note that Russell's paradox does not arise in this primitive system, because a strip number's "element" (we are free to avoid that loaded word if we choose) strips are always below it.
Now note that the top number has an S on the extreme left -- rather than a 0 -- such that S3 ⇒ S2 ⇒ S1 ⇒ 0 (where the sub numbers are only for our immediate convenience and have no intrinsic meaning; we could as well use prime marks or arbitrary names, as in Sdog ⇒ Sstarship.
In any case we may, only if we so desire, name the entire graphic above as a "set" or "proto-set." Similarly, we may so name each sub-graphic that occurs when we erase a top strip. Obviously, this parallels the usual set succession rules.
Though our naming these graphics "sets" is somewhat user-friendly, it is plainly unnecessary. We could as well name them with some random string of characters, say "xxjtdbn." The entire graph has the general name "xxjtdbn." Under it is another xxjtdbn, which differs from the other and so must take another name as a mark of distinction. In fact, every permissible graph must take some distinct name.
So for the entire graph we have "xxjtdbn." For the "next" sub-graph, we have "agbfsaf." For the one below that, we have "dtdmitg." And for the "0" strip we have "zbhikeb."
We are expected to know that each name applies to a specific strip and thus is either a successor or a predecessor or both. So then, we don't really need to employ the abstract concept of "set" (though we are employing abstract Euclidean axioms and a couple of other axioms).
Now if we write, for example,
(2x ∈ X)P
we seem to be saying that the more "advanced" set definitions are in force for "X" and so "2x ∈ X" is not a legitimate quantifier of the assertion (=proposition) P.
It is true that we are not done with our quantifier design. We are saying that "2" is a name given our graphic that is also known as "agbfsaf." We accept that there may be objects of some sort that go by the generic name "x." We must be able to establish a 1-1 correspondence between our agbfsaf graphic and any x's.
That is, we must be able to draw a single line (at least notionally) between every strip of agbfsaf, except the 0 strip, and one and only one of the x's. If that 1-1 correspondence is exact -- no unconnected ends -- then we may write (2!x ∈ X)P, which tells us that there are exactly 2 x's that apply to P "truthfully."
We have implied in our notation that there is a class X of which the x's are members. This isn't quite necessary. We may just say that X is a name for various x's. We might even say that we have a simple pairing system (aka "binary relation," though we must beware this terminology as probably anachronistic) such that xo,X, meaning that x may vary but that X may not.
Now suppose we wish to talk about equality, as in P means "x = x."
We may write (∀ x)(x = x), (∃ x)(x = x) and ~(∀ x)(x = x) in standard form.
We interrupt here to deny that the ∀ quantifier must be taken to require a set. Instead of using the concept "set," we say that there exists some formation of triples y,p,Y, such that y varies but p and Y do not. Y is associated with distinct y's.
Also, we are prissy about the words "all," "any" and "each." If a set or class is thought to be strictly implied by the word "all," then we disavow that word. Rather the ∀ quantifier is to be read as meaning that "any" y may be paired with (p,Y). That is, we say that we may select a y arbitrarily and must find that it has the name Y and the property p under consideration.[1]
Though the word "each" (="every") may connote "consecutive order" in the succession operation described above, it is neater for our purposes to use only the word "any" in association with the all quantifier.
Now if we mean by x one of the graphic "numbers," each graph shows a 1-1 correspondence with a copy of itself and so we can establish the first two statements as true and the last as false. If we mean that x represents arbitrary ZFC or NBG class theoretic numbers, we can still use the correspondence test (though we need not deal with indefinite unbounded algorithms -- loosely dubbed "infinity").
Further, we may use the correspondence test for, say, the graphic named "2." If we find that we can draw single lines connecting strips of "2" with objects known as x's among NBG or ZFC "numbers," then we can say 2x(x = x). That means that graphic "2" is 1-1 with some part of ZFC or NBG. Or, we would normally say, "there are at least 2 x's in NBG or ZFC that are equal to themselves (where "self" = "duplicate")," as opposed to 2!x(x = x), which would normally be verbalized as "there are exactly 2 x's in NBG or ZFC that are equal to themselves (or their duplicates). The last statement can only be true if we specify what x's we are talking about.
I concede that in these last few paragraphs I mix apples and oranges. Why would we need the graphic "numbers" if we already have ZFC or NBG? But, we do have proof of principle -- that much can be done with these successor graphics, or what some might term pseudo-sets.
1. In his book Logic for Mathematicians (Chelsea Publishing 1978, McGraw Hill 1953), J. Barkley Rosser cautions against the word any. "Sometimes any means each and sometimes it means some. Thus, sometimes "for any x..." means (x) and sometimes "for any x..." means (Ex)."
¶ [Note: (x) is an old-fashioned way to denote ∀x and (Ex) is Rosser's way of denoting ∃x.]
¶ After giving an ambiguous example, Rosser says, "If one wishes to be sure that one will be understood, one should never use any in a place where either each or some can be used. For instance, "I will not violate any law." The statement "I will not violate each law" has quite a different meaning, and the statement, "I will not violate some law" might be interpreted to mean that there is a particular law which I am determined not to violate.
¶ "Nonetheless," says Rosser, "many writers use any in places where each or some would be preferable."
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