It is my hope you someday find the courage to face reality and acknowledge that there are not gods. I am deeply sad that a person possessing a fine intellect such as your mind appears would purposefully turn their back on reason and rational philosophy and choose to become a Christian. The vile filth of Christianity and the vast evil it has wrought on western civilization sickens and disgusts rational reasoning people. If Yahweh does exist, I certainly would not want to continue with whatever this that we take for reality may then be, for if Yahweh exists the primacy of existence is false. In that case there is no fixed reality and this is some sort of sick illusion such as postulated by Descartes and the primacy of consciousness mystics. Nothingness or Hell would be preferable to being a slave to the monster before which you crawl on your belly, prostrating yourself, and worshiping what is arguably the most evil character in all of fictional literature while surrendering your moral autonomy. Shame on you for crouching down and licking the imaginary hand of a heinous delusion. But luckily it is such a remote impossibility that Yahweh might exist that I need not be concerned, for your god is a lie, and your religion is contemptible nonsense.

I erred in asserting that a Set was the rule by which elements were segregated into groupings. Although I distinctly remember the instructor in class making that point. Wikipedia defines a set as a group of elements. The remainder of the article fails to discuss the disposition of any rules used to define the set grouping. The role of rules used to perform segregation of elements seems to be generally dismissed. That is a mistake. For it leaves open the door to the insidious Analytic-Synthetic dichotomy fallacy. But first there are reasons why a group of an indefinitely and continuously increasing quantity of elements consisting of either of all even or all odd integers cannot have a certain identify.

I previously wrote:

*”Rules are finite even if the magnitude of the quantity of elements categorized by the rule can be counted forever. Measurement criteria used to discriminate differences and similarities between elements that are assigned to rules that are used to categorize groupings of things, ideas, concepts, or notions into sets are finite. Rules may apply to an infinite quantity, yet rules are finite.”*

The above observation is salient and cogent to the proposition that an infinite Set can have no specific identity because the rules used to instill meaning into an algorithm that is in turn used to segregate numbers from their native domain on the Real Number Line into an infinite Set as an ongoing process that continues for all time is not part of that Set. This means the definition of Even or Odd are not part or the infinite Set of all evens or odds.

In my prior missive, I further noted that:

*”Additionally, if we consider the set of all odd or even numbers and ask which number is the collective whole ensemble of all even or odd numbers, there can be no specific answer because the whole assemblage of elements has no identity.”*

Likewise this observation is cogent and salient. Consider the finite set {a,b,c}. What single letter of the alphabet is the set? As an ensemble the set is not the same as the identity of any one of its members. The set is the union, U, of all its proper subsets, but union, U, is a rule algorithm and is not part of the set {a,b,c,}. Consequently union, U, cannot be the identity of {a,b,c}. {a,b,c} could be thought of as the intersection, I, of two other set that each contain a, b, c in common, but intersection, I, is a rule algorithm and is not part of the set {a,b,c,}. The question, what single letter of the alphabet is the set?, cannot be answered. We can say that {a,b,c} is a and b and c, but we cannot say {a,b,c} is any certain letter. What can be done, however, is apply an arbitrary label to the set grouping that entails it is the group of the first three letters of the alphabet.

Consider a bowl containing three marbles, one blue, one red, one yellow. It can be asked, what color is the group of three marbles?, but no certain specific answer can be forthcoming as the group has no specific color. The group does have three marbles and each one has a specific color, but the other two do likewise have their own specific color. The group has all the properties of all its elements, so the group is red, yellow, and blue simultaneously. But the set {blue marble, red marble, yellow marble} also has the property of marble. If the marbles are composed of glass, then the group also has the property of glass. Marbles are spheroidal; the set {blue marble, red marble, yellow marble} would then also have the property of being spheroidal. By which property shall we identify the set? If we were being truthful in the sense laid out in ITOE

*”Truth is the product of the recognition (i.e., identification) of the facts of reality. Man identifies and integrates the facts of reality by means of concepts. He retains concepts in his mind by means of definitions. He organizes concepts into propositions—and the truth or falsehood of his propositions rests, not only on their relation to the facts he asserts, but also on the truth or falsehood of the definitions of the concepts he uses to assert them, which rests on the truth or falsehood of his designations of essential characteristics.”*- p.63

How do we decide which essential characteristic is most appropriate to defining what color is the set {blue marble, red marble, yellow marble}? Is the set not defined by all of its properties that are inherited from its elements? Its seems clear that is the case. Thus {blue marble, red marble, yellow marble} is blue, red, and yellow. What if instead of three marbles, there was and infinite quantity of marbles of all different varying shades and hues? What color would then the set {∞ x all different colored marbles} be? Plainly, the color would then be indescribable for there is no such entity as actual infinity. The fantasies of math geeks notwithstanding. I posted Peikoff's take on the matter of infinity previously and its worth revisiting.

*‘Infinite’ do not mean large; it means larger than any specific quantity, i.e.: of no specific quantity. An infinite quantity would be a quantity without identity. But A is A. Every entity, accordingly, is finite; it is limited in the number of its qualities and in their extent; this applies to the universe as well. As Aristotle was the first to observe, the concept of ‘infinity’ denotes merely a potentiality of indefinite addition or subdivision. For example, one can*

continually subdivide a line; but however many segments one has reached at a given point, there are only that many and no more. The actual is always finite.- “Objectivism: The Philosphy of Ayn Rand”, p.31, by Leonard Peikoff

continually subdivide a line; but however many segments one has reached at a given point, there are only that many and no more. The actual is always finite.

And I also pointed out George H. Smith's brilliancy on this issue.

*To exist is to exist as something. To be something is to have a specific nature. That is to have a particular identity. The Laws of Identity A=A and Non-Contradiction A =/= ¬A entail that any ontological being must posses specific determinate characteristics. To have such characteristics is a consequence of being part of nature ..... Having specific determinate characteristics imposes limits, and those limits would restrict the capacities of the .... being. Such restriction then renders the .... being subject to the causal relationships that denote the uniformity of nature in actual existence ....*- “Atheism: The Case Against God.”, p.41 (paraphrasing), by George H. Smith

To be subject to causality is to operate in harmony with the nature of existence. Causality is the law of identity applied to action. All actions are caused by entities. The nature of an action is caused and determined by the nature of the entities that act; a thing cannot act in contradiction to its nature. The algorithms the segregate odd and even numbers from their native domain on the real number line filters numbers that are an integer of the form n = 2k + 1, where k is an integer into the odd Set and filters numbers having the form n = 2k where k is an integer into the even Set. Casualty is recognized as applying to algorithm rules that categorize numbers into sets as well as to specific numbers, but not to an infinite quantity of numeric elements categorized by an algorithm rule into a set of numbers. For if infinite were to exist, they would not have specific natures. This is because a concept, {the set of all even or odd numbers} for instance, means the existents which it subsumes, including all their characteristics and properties.

The rule algorithm entity that operates in finite fashion to segregate integers into the even or odd Sets is not part of the set. The definition of even or odd is then not part of the set. Therefore that which is infinite has no specific identity, for the identity consists of all the properties of the set inherited from all of its elements and subsets. Your claim that the alleged infinite set of all evens or odds can have an identity is pure hogwash. You are simply parroting the math geeks who are fallaciously applying an arbitrary even or odd label to the sets in question while ignoring all of the set's other properties that go into making up its actual identity. That is Set Theory commits the Analytic-Synthetic dichotomy fallacy. I discuss this further at the end of the essay.

You, being a Christian mystic, may wish to appeal to a Universal form emanating from the transcendent realm of your god that confers evenness or oddness on the Set. Since the rule that segregates the evens or the odds into the Set is not part of the Set, then your unstated enthymeme of a Universal cannot apply to the members of the Set because the definition rule EVEN or ODD is not part of the Set. Additionally, per Grupp (as I previously mentioned and provided a link to his paper) a relation cannot occur between that which is wholly spatial and temporal ie: existence and that which is non-spatial and A-temporal, ie: transcendence. No Universal form of EVEN-ness or ODD-ness can influence the hypothetical groupings. It is an arbitrary action to assign a label of even or odd to the Set, as the most essential characteristic of the elements in the set are their numeric magnitude.

You may wish to protest that the individual numbers within the ensemble each have the property of EVEN or ODD. But this will not do because a thing is all that it is. The Set is all its elements, then it must also be all of its element's properties and all of its subsets. Each number can be considered a proper subset of the main set. Not only is then each number different, but it is composed of an infinite number of fractional rational and irrational divisions and summation sequences each of which is a proper subset of its integer. All of the numbers within the main Set are also members of other Sets that can be defined and that are proper subsets of the main Set. An infinite number of encapsulated Sets occurs within any give sequence of numbers, and since Sets are defined as objects by Set theory, they must exist. If they exist they have properties even if we do not know what they are. Thus there is a vast multiplicity of Sets with their own properties within the main Set. The main Set's identity must include all its properties and those of all its proper subsets and not just EVEN or ODD. That math geeks wrongfully and arbitrarily apply the EVEN or ODD rule labels to these imaginary structures does not actually identify the Set any more than calling me Robert identifies me. I am all that I am and the arbitrarily applied label stuck on me by my parents is not my identity. It is my label. The Sets identity is like mixing all colors of the marbles together. What color then results? Why, no color at all, for black is not a color. What magnitude of quantity is the ensemble of the main Set?, why no specific magnitude of quantity at all for all numbers taken together in a group cannot be any certain number.

Consider Pure Sets. A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty or Null Set is a non-empty pure Set. If the Set containing the Null Set is non-empty, then the Null set must be something according to Set theory. But the Null Set is the grouping of its members. In the Null set resides its member, nothingness. A grouping of nothingness is nothingness. Yet Set Theory reifies nothingness into somethingness. Let W = a Pure Set that is an sequential encapsulation of an infinite quantity of instances of Null Sets like Russian Babushka dolls. Each order of successively encapsulated Null Sets is assigned a ranking with its ordinal number. As the sets are encapsulated each set will contain its own powerset. The main Set will then have cardinality greater than Aleph 0. Now imagine, (F), a one-to-one injective function correspondence between W and the Set of all evens. Since W contains its own powerset, it has 2^(Aleph 0) more elements than does a non-Babushka doll-sequentially-encapsulated Pure Set containing an Aleph 0 quantity of Null Sets. After (F) is applied, then the remaining |2^(Aleph 0)- Aleph 0| Russian Babushka doll Null Sets in W will still be uncountable and still be equivalent to nothingness. Yet the remnant of W will have a greater quantity of elements than the Set of all Evens. This is absurd. How can nothingness be greater than infinity? It can't be. But it is indicative of a basic contradiction that renders the notion of an infinite Sets incomprehensible and incoherent and thus non-identifiable, but math geeks can arbitrarily apply those EVEN or ODD labels to the indeterminate groupings so stipulated by way of the Analytic-Synthetic Dichotomy fallacy.

Consider a Set of all even numbers {E} that results from a bijective functional mapping, f, such that each element is the product of {N}={2,4,6,....,n(sub ∞)} and (Aleph 0^(Aleph 0)) as follows: {E}=f(n)=n * (Aleph 0^(Aleph 0)). Each element in {E} is the product of one of the even natural numbers X (Aleph 0^(Aleph 0)). Plotted on a number line each element of {E} would be separated by (Aleph 0 raised to the (Aleph 0) exponential power) quantity of numeric points. The reciprocal of (Aleph 0^(Aleph 0)) or 1/(Aleph 0^(Aleph 0)) is the LIM of x as x→0 . The cardinality of the total number of elements of in {E} would then be the (LIM of x as x→0) * (The cardinality of the total number of elements of {N}). The cardinality of {E} then would be very much like zero, But a bijective functional mapping of a one-to-one correspondence between elements of {N} and {E} was established. Both sets are countable, but {E} has Aleph 0 cardinality while {N} has LIM of x as x→0 cardinality. Here is a definite relationship between Zero and ∞. But zero is nothingness, and a definite relationship between nothingness and somethingness cannot occur. (See Grupp.) The concept of infinity (or eternity) is incoherent and incomprehensible and cannot actually exist.

Whoever asserts that the fantasy of an infinite set (Yes, it is a fantasy as there can be no actual infinity.) of even or odd numbers to have the identity of even or odd is committing the Analytic-Synthetic Dichotomy fallacy.

*An analytic proposition is defined as one which can be validated merely by an analysis of the meaning of its constituent concepts. The critical question is: What is included in “the meaning of a concept”? Does a concept mean the existents which it subsumes, including all their characteristics? Or does it mean only certain aspects of these existents, designating some of their characteristics but excluding others?*

The latter viewpoint is fundamental to every version of the analytic-synthetic dichotomy. The advocates of this dichotomy divide the characteristics of the existents subsumed under a concept into two groups: those which are included in the meaning of the concept, and those—the great majority—which, they claim, are excluded from its meaning. The dichotomy among propositions follows directly. If a proposition links the “included” characteristics with the concept, it can be validated merely by an “analysis” of the concept; if it links the “excluded” characteristics with the concept, it represents an act of “synthesis.”- ITOE, p.127 Peikoff

The latter viewpoint is fundamental to every version of the analytic-synthetic dichotomy. The advocates of this dichotomy divide the characteristics of the existents subsumed under a concept into two groups: those which are included in the meaning of the concept, and those—the great majority—which, they claim, are excluded from its meaning. The dichotomy among propositions follows directly. If a proposition links the “included” characteristics with the concept, it can be validated merely by an “analysis” of the concept; if it links the “excluded” characteristics with the concept, it represents an act of “synthesis.”

*The Objectivist theory of concepts undercuts the theory of the analytic-synthetic dichotomy at its root .... Since a concept is an integration of units, it has no content or meaning apart from its units. The meaning of a concept consists of the units—the existents—which it integrates, including all the characteristics of these units. Observe that concepts mean existents, not arbitrarily selected portions of existents. There is no basis whatever—neither metaphysical nor epistemological, neither in the nature of reality nor of a conceptual consciousness—for a division of the characteristics of a concept’s units into two groups, one of which is excluded from the concept’s meaning ....The fact that certain characteristics are, at a given time, unknown to man, does not indicate that these characteristics are excluded from the entity—or from the concept. A is A; existents are what they are, independent of the state of human knowledge; and a concept means the existents which it integrates. Thus, a concept subsumes and includes all the characteristics of its referents, known and not-yet-known.*- ITOE, p.131 Peikoff

On pages 98-101 of Introduction to Objectivist Epistemology Expanded 2nd Edition, Meridian Penguin Books, April 1990, Leonard Peikoff demonstrate how the Objectivist theory of concepts defangs and neuters the Analytic-Synthetic Dichotomy. By a fine example of reasoning Peikoff notes the following:

*I)Metaphysically, and entity is: all of the things which it is. Each of its characteristics has the same metaphysical status: each constitutes a part of the entity's identity.*

II)Epistemologically, all the characteristics of the entities subsumed under a concept are discovered by the same basic method: by observation of these entities.

III)... a concept subsumes and includes all the characteristics of its referents, known and not-yet-known.

IV)....a concept is an open-end classification which includes the yet-to-be discovered characteristics of a given group of existents. All of man's knowledge rest on that fact.

V)Whatever is true of the entity, is meant by the concept.

VI)It follows that there are no grounds on which to distinguish “analytic” from “synthetic” propositions. Whether on state that “A man is a rational animal” or that “A man has only two eyes” - in both cases, the predicated characteristics are true of man and are, therefore, included in the concept “man”. The meaning of the first statement is: “A certain type of entity , including all its characteristics (among which are rationality and animality) is: a rational animal.” The meaning of the second is: “A certain type of entity, including all of its characteristics (among which is the possession of only two eyes) has: only two eyes.” Each of these statements is an instance of the Law of Identity; each is a “tautology”: to deny either is to contradict the meaning of the concept “man,” and thus to endorse a self-contradiction.

II)Epistemologically, all the characteristics of the entities subsumed under a concept are discovered by the same basic method: by observation of these entities.

III)... a concept subsumes and includes all the characteristics of its referents, known and not-yet-known.

IV)....a concept is an open-end classification which includes the yet-to-be discovered characteristics of a given group of existents. All of man's knowledge rest on that fact.

V)Whatever is true of the entity, is meant by the concept.

VI)It follows that there are no grounds on which to distinguish “analytic” from “synthetic” propositions. Whether on state that “A man is a rational animal” or that “A man has only two eyes” - in both cases, the predicated characteristics are true of man and are, therefore, included in the concept “man”. The meaning of the first statement is: “A certain type of entity , including all its characteristics (among which are rationality and animality) is: a rational animal.” The meaning of the second is: “A certain type of entity, including all of its characteristics (among which is the possession of only two eyes) has: only two eyes.” Each of these statements is an instance of the Law of Identity; each is a “tautology”: to deny either is to contradict the meaning of the concept “man,” and thus to endorse a self-contradiction.

When someone labels a so-called infinite Set defined by the rules EVEN or ODD they are denying that the sequence of numbers is all that it actually is. Placing a label on something that is incomprehensible, incoherent and which does not actually exist is to deny (I-VI). By assuming that a Universal transcendence form of Even or Odd somehow crosses from a transcendence to our reality, the delusional believer violates (III). No certain identity can be ascertained from an infinite set. An arbitrarily applied label can be suck on it to give an impression of an identity, but such labels are no more identity than is my name, my identity. But labeling an infinite set with one of its characteristics while ignoring the remaining characteristics is constructively a lie, a falsehood, a deceit.

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