Closed Loop Interval Ontology
     CLOSED LOOP INTERVAL ONTOLOGY
       The Digital Integration of Conceptual Form

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The closed loop ensemble contains
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Dimensions of set theory

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Set
Dimensions of set theory

The foundational concept of mathematics is "set".

In mathematics, a set is a collection of distinct elements. The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets. Two sets are equal if and only if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. The subject called set theory is part of the foundations of mathematics.

Set theory (all)
Set theory and geometry
Set theory (naïve)
Set theory (axiomatic)
Set theory (class)
Ordered set
Partially ordered set
Set theory and dimensionality
Similarities and differences
Linnaeus on similarities and differences
Set and boundary
Set, venn diagram, dimensionality, boundary
Abstract object, concrete object
Russell's Paradox
Binary relationship (math)

Set theory (all)
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A set is a collection of objects

Reference
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naïve set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known.

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

URL
https://en.wikipedia.org/wiki/Set_theory

Set theory and geometry
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We want to explore deriving the concept of set and "members of a set" from geometry -- from what used to be Euclid, to what we are now exploring as "digital geometry"

Euclid defines a "line" -- we want to strengthen that definition

Lines define the boundaries of set and what is contained within set -- the boundaries between the objects -- the members of the set -- when they are ordered

**

And of course -- there are Venn diagrams -- circular boundaries around things they contain

Set theory (naïve)
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Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.[1] Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.[2]

Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments.

**

Some important sets

There are some ubiquitous sets for which the notation is almost universal. Some of these are listed below. In the list, a, b, and c refer to natural numbers, and r and s are real numbers.

Natural numbers are used for counting. A blackboard bold capital N often represents this set.

Integers appear as solutions for x in equations like x + a = b. A blackboard bold capital Z often represents this set (from the German Zahlen, meaning numbers).

Rational numbers appear as solutions to equations like a + bx = c. A blackboard bold capital Q often represents this set (for quotient, because R is used for the set of real numbers).

Algebraic numbers appear as solutions to polynomial equations (with integer coefficients) and may involve radicals and certain other irrational numbers. A Q with an overline often represents this set. The overline denotes the operation of algebraic closure.

Real numbers represent the "real line" and include all numbers that can be approximated by rationals. These numbers may be rational or algebraic but may also be transcendental numbers, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital R often represents this set.

Reference
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.[1] Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.[2]

Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments.

URL
https://en.wikipedia.org/wiki/Naive_set_theory

Set theory (axiomatic)
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Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox.

Axiomatic set theory was originally devised to rid set theory of such paradoxes.

URL
https://en.wikipedia.org/wiki/Set_theory#Axiomatic_set_theory

Set theory (class)
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In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.

In Quine's set-theoretical writing, the phrase "ultimate class" is often used instead of the phrase "proper class" emphasising that in the systems he considers, certain classes cannot be members, and are thus the final term in any membership chain to which they belong.

Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps rather take place without considering that certain classes can fail to be sets.

URL
https://en.wikipedia.org/wiki/Class_(set_theory)

Ordered set
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In mathematics, a total order, simple order,[1] linear order, connex order,[2] or full order[3] is a binary relation on some set {\displaystyle X}X, which is antisymmetric, transitive, and a connex relation. A set paired with a total order is called a chain,[4] a totally ordered set,[4] a simply ordered set,[1] a linearly ordered set,[2][4] or a loset.[5][6]

Formally, a binary relation {\displaystyle \leq }\leq is a total order on a set {\displaystyle X}X if the following statements hold for all {\displaystyle a,b}a,b and {\displaystyle c}c in {\displaystyle X}X:

Antisymmetry: If {\displaystyle a\leq b}a\leq b and {\displaystyle b\leq a}{\displaystyle b\leq a} then {\displaystyle a=b}a=b; Transitivity: If {\displaystyle a\leq b}a\leq b and {\displaystyle b\leq c}{\displaystyle b\leq c} then {\displaystyle a\leq c}{\displaystyle a\leq c}; Connexity: {\displaystyle a\leq b}a\leq b or {\displaystyle b\leq a}{\displaystyle b\leq a}.

Antisymmetry eliminates uncertain cases when both {\displaystyle a}a precedes {\displaystyle b}b and {\displaystyle b}b precedes {\displaystyle a}a.[7]:325 A relation having the connex property means that any pair of elements in the set of the relation are comparable under the relation. This also means that the set can be diagrammed as a line of elements, giving it the name linear.[7]:330 The connex property also implies reflexivity, i.e., a ? a. Therefore, a total order is also a (special case of a) partial order, as, for a partial order, the connex property is replaced by the weaker reflexivity property. An extension of a given partial order to a total order is called a linear extension of that partial order.

URL
https://en.wikipedia.org/wiki/Total_order

Partially ordered set
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What we seem to be seeing in the notion of a "partially ordered set" is that the "pure linear order" proceeds in descending/vertical levels, but there may be branching between the levels, which extends in a "partially horizontal" level.

Another interesting topic is the subject of "ordered by inclusion"

Which would seem to mean that A contains B and B contains C, therefore there is a linear ordering of these three elements, A, B, C.

Reference
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.

Formally, a partial order is any binary relation that is reflexive (each element is comparable to itself), antisymmetric (no two different elements precede each other), and transitive (the start of a chain of precedence relations must precede the end of the chain).

One familiar example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.

A poset can be visualized through its Hasse diagram, which depicts the ordering relation.[1]

URL
https://en.wikipedia.org/wiki/Partially_ordered_set

Set theory and dimensionality
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This is where we want to go -- to show that all elements of classical set theory can be defined in terms of dimensionality

  • A set itself is a dimension -- because all objects in the set are not exactly identical -- or if they are, they can be ordered by the integers -- if they are not, they can be ordered by their differences
  • The objects in (of) the set can be defined in terms of dimensionality
    • Apples
    • Oranges
  • What is the differences between a set and a class?

Similarities and differences
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Following this reference and the leading quote from Linnaeus, it seems like what it important is that perceptions begin with the recognition of differences, which then highlight similarities. This was a strong theme in the early formation of synthetic dimensionality.

We want to define similarities as similar or identical values in similar or identical dimensions.

Defining similarity in a scientific or precise analytic way has proven difficult, and there is no wide-spread agreement as to what it means.

Maybe we get some clues from concepts such as metaphor, simile and analogy.

Reference
Conceptions of similarity

Conceptions of similarity give an account of similarity and its degrees on a metaphysical level. The simplest view, though not very popular, sees resemblance as a fundamental aspect of reality that cannot be reduced to other aspects.[3][10] The more common view is that the similarity between two things is determined by other facts, for example, by the properties they share, by their qualitative distance or by the existence of certain transformations between them.[5][11] These conceptions analyze resemblance in terms of other aspects instead of treating it as a fundamental relation.

Numerical

The numerical conception holds that the degree of similarity between objects is determined by the number of properties they have in common.[12] On the most basic version of this view, the degree of similarity is identical to this number. For example, "[i]f the properties of peas in a pod were just greenness, roundness and yuckiness ... then their degree of similarity would be three".[13] Two things need to share at least one property to be considered similar. They resemble each other exactly if they have all their properties in common. This is also known as qualitative identity or indiscernibility. For the numerical conception of similarity to work, it is important that only properties relevant to resemblance are taken into account, sometimes referred to as sparse properties in contrast to abundant properties.[13][14] Quantitative properties, like temperature or mass, which occur in degrees, pose another problem for the numerical conception.[3] The reason for this is that e.g. a body with 40 °C resembles another body with 41 °C even though the two bodies do not have their temperature in common.

Metric

The problem of quantitative properties is better handled by the metric conception of similarity, which posits that there are certain dimensions of similarity concerning different respects, e.g. color, shape or weight, which constitute the axes of one unified metric space.[13][3] This can be visualized in analogy to three-dimensional physical space, the axes of which are usually labeled with x, y and z.[8] In both the qualitative and the physical metric space, the total distance is determined by the relative distances within each axis. The metric space thus constitutes a manner of aggregating various respective degrees of similarity into one overall degree of similarity.[9][8] The corresponding function is sometimes referred to as a similarity measure. One problem with this outlook is that it is questionable whether the different respects are commensurable with each other in the sense that an increase in one type can make up for a lack in another type.[9] Even if this should be allowed, there is still the question of how to determine the factor of correlation between degrees of different respects.[3] Any such factor would seem to be artificial,[8] as can be seen, for example, when considering possible responses to the following case: "[l]et one person resemble you more closely, overall, than someone else does. And let him become a bit less like you in respect of his weight by gaining a little. Now answer these questions: How much warmer or cooler should he become to restore the original overall comparison? How much more similar in respect of his height?"[9] This problem does not arise for physical distance, which involves commensurable dimensions and which can be kept constant, for example, by moving the right amount north or south, after having moved a certain distance to the west.[9][8] Another objection to the metric conception of similarity comes from empirical research suggesting that similarity judgments do not obey the axioms of metric space. For example, people are more likely to accept that "North Korea is similar to China" than that "China is similar to North Korea", thereby denying the axiom of symmetry.[12][3]

Transformation

Another way to define similarity, best known from geometry, is in terms of transformations. According to this definition, two objects are similar if there exists a certain type of transformation that translates one object into the other object while leaving certain properties essential for similarity intact.[11][5] For example, in geometry, two triangles are similar if there is a transformation, involving nothing but scaling, rotating, displacement and reflection, which maps one triangle onto the other. The property kept intact by these transformations concerns the angles of the two triangles.[11]

Exact similarity and identity

Identity is the relation each thing bears only to itself.[15] Both identity and exact similarity or indiscernibility are expressed by the word "same".[16][17] For example, consider two children with the same bicycles engaged in a race while their mother is watching. The two children have the same bicycle in one sense (exact similarity) and the same mother in another sense (identity).[16] The two senses of sameness are linked by two principles: the principle of indiscernibility of identicals and the principle of identity of indiscernibles. The principle of indiscernibility of identicals is uncontroversial and states that if two entities are identical with each other then they exactly resemble each other.[17] The principle of identity of indiscernibles, on the other hand, is more controversial in making the converse claim that if two entities exactly resemble each other then they must be identical.[17] This entails that "no two distinct things exactly resemble each other".[18] A well-known counterexample comes from Max Black, who describes a symmetrical universe consisting of only two spheres with the same features.[19] Black argues that the two spheres are indiscernible but not identical, thereby constituting a violation of the principle of identity of indiscernibles.[20]

Applications in philosophy

Problem of universals

The problem of universals is the problem to explain how different objects can have a feature in common and thereby resemble each other in this respect, for example, how water and oil can share the feature of being liquid.[21][22] The realist solution posits an underlying universal that is instantiated by both objects and thus grounds their similarity.[16] This is rejected by nominalists, who deny the existence of universals. Of special interest to the concept of similarity is the position known as resemblance nominalism, which treats resemblance between objects as a fundamental fact.[22][16] So on this view, two objects have a feature in common because they resemble each other, not the other way round, as is commonly held.[23] This way, the problem of universals is solved without the need of positing shared universals.[22] One objection to this solution is that it fails to distinguish between coextensive properties. Coextensive properties are different properties that always come together, like having a heart and having a kidney. But in resemblance nominalism, they are treated as one property since all their bearers belong to the same resemblance class.[24] Another counter-argument is that this approach does not fully solve the problem of universals since it seemingly introduces a new universal: resemblance itself.[22][3]

Counterfactuals

Counterfactuals are sentences that express what would have been true under different circumstances, for example, "[i]f Richard Nixon had pushed the button, there would have been a nuclear war".[25] Theories of counterfactuals try to determine the conditions under which counterfactuals are true or false. The most well-known approach, due to Robert Stalnaker and David Lewis, proposes to analyze counterfactuals in terms of similarity between possible worlds.[7][26] A possible world is a way things could have been. According to the Stalnaker-Lewis-account, the antecedent or the if-clause picks out one possible world, in the example above, the world in which Nixon pushed the button. The counterfactual is true if the consequent or the then-clause is true in the selected possible world.[26][7] The problem with the account sketched so far is that there are various possible worlds that could be picked out by the antecedent. Lewis proposes that the problem is solved through overall similarity: only the possible world most similar to the actual world is selected.[25] A "system of weights" in the form of a set of criteria is to guide us in assessing the degree of similarity between possible worlds.[7] For example, avoiding widespread violations of the laws of nature ("big miracles") is considered an important factor for similarity while proximity in particular facts has little impact.[7] One objection to Lewis' approach is that the proposed system of weights captures not so much our intuition concerning similarity between worlds but instead aims to be consonant with our counterfactual intuitions.[27] But considered purely in terms of similarity, the most similar world in the example above is arguably the world in which Nixon pushes the button, nothing happens and history continues just like it actually did.[27]

Depiction

Depiction is the relation that pictures bear to the things they represent, for example, the relation between a photograph of Albert Einstein and Einstein himself. Theories of depiction aim to explain how pictures are able to refer.[28] The traditional account, originally suggested by Plato, explains depiction in terms of mimesis or similarity.[29][30] So the photograph depicts Einstein because it resembles him in respect to shape and color. In this regard, pictures are different from linguistic signs, which are arbitrarily related to their referents for most part.[28][30] Pictures can indirectly represent abstract concepts, like God or love, by resembling concrete things, like a bearded man or a heart, which we associate with the abstract concept in question.[29] Despite their intuitive appeal, resemblance-accounts of depiction face various problems. One problem comes from the fact that similarity is a symmetric relation, so if a is similar to b then b has to be similar to a.[28] But Einstein does not depict his photograph despite being similar to it. Another problem comes from the fact that non-existing things, like dragons, can be depicted. So a picture of a dragon shows a dragon even though there are no dragons that could be similar to the picture.[28][30] Defenders of resemblance-theories try to avoid these counter-examples by moving to more sophisticated formulations involving other concepts beside resemblance.[29]

Argument from analogy

An analogy is a comparison between two objects based on similarity.[31] Arguments from analogy involve inferences from information about a known object (the source) to the features of an unknown object (the target) based on similarity between the two objects.[32] Arguments from analogy have the following form: a is similar to b and a has feature F, therefore b probably also has feature F.[31][33] Using this scheme, it is possible to infer from the similarity between rats (a) and humans (b) and from the fact that birth control pills affect the brain development (F) of rats that they may also affect the brain development of humans.[34] Arguments from analogy are defeasible: they make their conclusion rationally compelling but do not ensure its truth.[35] The strength of such arguments depends, among other things, on the degree of similarity between the source and the target and on the relevance of this similarity to the inferred feature.[34] Important arguments from analogy within philosophy include the argument from design (the universe resembles a machine and machines have intelligent designers, therefore the universe has an intelligent designer) and the argument from analogy concerning the existence of other minds (my body is similar to other human bodies and I have a mind, therefore they also have minds).[32][36][37][38]

Family resemblance

The term family resemblance refers to Ludwig Wittgenstein's idea that certain concepts cannot be defined in terms of necessary and sufficient conditions which refer to essential features shared by all examples.[39][40] Instead, the use of one concept for all its cases is justified by resemblance relations based on their shared features. These relations form "a network of overlapping but discontinuous similarities, like the fibres in a rope".[40] One of Wittgenstein's favorite examples is the concept of games, which includes card games, board games, ball games, etc. Different games share various features with each other, like being amusing, involving winning and losing, depending on skill or luck, etc.[41] According to Wittgenstein, to be a game is to be sufficiently similar to other games even though there are no properties essential to every game.[39] These considerations threaten to render traditional attempts of discovering analytic definitions futile, such as for concepts like proposition, name, number, proof or language.[40] Prototype theory is formulated based on these insights. It holds that whether an entity belongs to a conceptual category is determined by how close or similar this entity is to the prototype or exemplar of this concept.[42][43]

URL
https://en.wikipedia.org/wiki/Similarity_(philosophy)

Linnaeus on similarities and differences
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From http://originresearch.com

All the real knowledge which we possess depends on methods by which we distinguish the similar from the dissimilar. The greater the number of natural distinctions this method comprehends the clearer becomes our idea of things. The more numerous the objects which employ our attention the more difficult it becomes to form such a method, and the more necessary. -- Carolus Linnaeus, Genera Plantarum, 1737

We are exploring a new algebraic theory of knowledge representation based on the isomorphism of "dimension" and "ordered class". A "synthetic dimension" is defined recursively, as an ordered class of variables, themselves described in synthetic dimensions. All abstract classes can be assembled from lowest-level empirical measurements through this method

All empirical measurement is defined by values in dimensions. All abstract classes are defined by composite assemblies of dimensions, and an ordered class can be shown to be algebraically isomorphic to a dimension. Thus, any high-level composite abstraction can be determinately defined bottom-up from lowest-level empirical/dimensional measurements. This approach creates a linearly recursive cascade across ascending levels of abstraction, through which any concept, category, classification, or data structure can be exactly defined, to any desired degree of accuracy/error tolerance (number of decimal places in measurement).

Synthetic Dimensionality is, at the same time, a model of the foundations of mathematics (real number line), an intuitive theory of natural language semantics, an algebraic theory of classification, and a general theory of all conceptual structure. Any idea, concept, mental model, or "information structure" can be constructed in perfect detail with synthetic dimensions.

Set and boundary
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We want to generalize the notion of set in terms of dimensionality and boundary values

Set, venn diagram, dimensionality, boundary
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Set, venn diagram, dimensionality, boundary

Define the concept of set in terms of boundary and its objects in terms of dimensionality

(abstract object, concrete object)

At a minimum, a set has a boundary -- things IN the set, and things NOT IN the set

Can that be specified by dimensionality?

Yes, as long as the objects are defined in terms of dimensionality

Abstract objects are defined in terms of dimensionality, and then a mapping/correlation is established by the abstract object (the word, the symbol) and the concrete object ("the actual thing" it is pointing to))

Set
Venn diagrams

Abstract object, concrete object
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Most of the mathematics we are exploring has to do with abstract objects -- by which we mean some symbolic representation -- usually in an alphabet -- and actually existing in some medium -- such as words written on printed on a piece of paper.

Is a generalization or "universal" an abstract object? I would say yes, in that its only concrete actuality/reality is that it points to a class of specific objects -- which in some cases might also be abstract or universal

Reference
In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, human beings and planets while things like numbers, sets and propositions are abstract objects.[1] There is no general consensus as to what the characteristic marks of concreteness and abstractness are. Popular suggestions include defining the distinction in terms of the difference between (1) existence inside or outside space-time, (2) having causes and effects or not, (3) having contingent or necessary existence, (4) being particular or universal and (5) belonging to either the physical or the mental realm or to neither.[2][3][4] Despite this diversity of views, there is broad agreement concerning most objects as to whether they are abstract or concrete.[1] So under most interpretations, all these views would agree that, for example, plants are concrete objects while numbers are abstract objects.

Abstract objects are most commonly used in philosophy and semantics. They are sometimes called abstracta in contrast to concreta. The term abstract object is said to have been coined by Willard Van Orman Quine.[5] Abstract object theory is a discipline that studies the nature and role of abstract objects. It holds that properties can be related to objects in two ways: through exemplification and through encoding. Concrete objects exemplify their properties while abstract objects merely encode them. This approach is also known as the dual copula strategy.[6]

The type–token distinction identifies physical objects that are tokens of a particular type of thing.[7] The "type" of which it is a part is in itself an abstract object. The abstract–concrete distinction is often introduced and initially understood in terms of paradigmatic examples of objects of each kind:

URL
https://en.wikipedia.org/wiki/Abstract_and_concrete

Russell's Paradox
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Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction.

Russell’s paradox

Bertrand Russell (1872-1970) was involved in an ambitious project to rewrite all the truths of mathematics in the language of sets. In fact, what he was trying to do was show that all of mathematics could be derived as the logical consequences of some basic principles using sets. At this time (around 1900), it was generally believed that any property of objects could define a set. For example, the property “x is a natural number between four and seven” defines the set {4, 5, 6, 7}. We could also write this set as {x| x is a natural number between 4 and 7}. What was believed was precisely that for any property “P” that you can think of, it made sense to talk about the set {x| x has property P}. Certainly, sets that consist of numbers make sense this way. But when you allow any objects in your sets, you can run into trouble.

Russell was the first one to notice this.

Russell’s insight was the following. First, it is possible for a set to be an element of itself. (Remember that elements are the objects which make up the set, e.g. the number 4 is an element of the set {4, 5, 6, 7}). An example of a set which is an element of itself is {x| x is a set and x has at least one element}.

This set contains itself, because it is a set with at least one element. Using this knowledge, Russell defined a special set, which we’ll call “R”. R is the set {x| x is a set and x is not an element of itself}.

Russell then asked: is R an element of the set R? Let’s think about this question.

If R is an element of R, that exactly means that it is an element of itself.

Which means that it can’t possibly be in R - by definition R is the collection of all sets which are not elements of themselves.

Since this option is impossible, we must agree that R is not an element of R.

But in that case, R is not an element of itself, so by definition it belongs to the collection of sets which are not elements of themselves. Uh-oh! [It is worth pausing a minute here and reassessing the situation on your own. Do you believe that R is an element of R or not? Neither? What’s the problem?]

Mathematicians and logicians thought for a while that the problem with the set R was going to undermine their whole project to do all mathematics in terms of set theory. Fortunately, they eventually came up with a technical solution that changes the way sets are constructed (slightly) and prevents R from being considered a set (whew!). This technical solution is why you sometimes see a set being described in the form {x ? X| x has property P}, where “X” is supposed to denote some other set, or “the universe” (whatever that means!) We won’t worry about that issue in our class, as most of our sets will be sets of numbers and these are always safe.

So did Russell succeed? Yes and no. The program to do all mathematics in terms of set theory was the birth of modern day logic, a rich and exciting area of mathematics. But ultimately, the logician Kurt G¨odel showed that the original goal of the project – to deduce all mathematics from some axioms (rules) concerning sets – was itself mathematically impossible!

For more details on this story, I recommend the excellent book Logicomix by Doxiades and Papadimitriou. You may be delighted to know that a) it’s a graphic novel and b) the Reg library has a copy.

Reference
In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901,[1][2] showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a contradiction. The same paradox had been discovered in 1899 by Ernst Zermelo[3] but he did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other members of the University of Göttingen. At the end of the 1890s Cantor himself had already realized that his definition would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter.[4]

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. Symbolically:

{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}{\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R In 1908, two ways of avoiding the paradox were proposed: Russell's type theory and the Zermelo set theory. Zermelo's axioms went well beyond Gottlob Frege's axioms of extensionality and unlimited set abstraction; as the first constructed axiomatic set theory, it evolved into the now-standard Zermelo–Fraenkel set theory (ZFC). The essential difference between Russell's and Zermelo's solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed, while Russell altered the logical language itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be first-order logic.[5]

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https://en.wikipedia.org/wiki/Russell%27s_paradox

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In mathematics (specifically set theory), a binary relation over sets X and Y is a subset of the Cartesian product X × Y; that is, it is a set of ordered pairs (x, y) consisting of elements x in X and y in Y.[1] It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1, ..., Xn, which is a subset of the Cartesian product X1 × ... × Xn.[1][2]

An example of a binary relation is the "divides" relation over the set of prime numbers {\displaystyle \mathbb {P} }\mathbb {P} and the set of integers {\displaystyle \mathbb {Z} }\mathbb {Z} , in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as ?4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.

Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:

the "is greater than", "is equal to", and "divides" relations in arithmetic; the "is congruent to" relation in geometry; the "is adjacent to" relation in graph theory; the "is orthogonal to" relation in linear algebra. A function may be defined as a special kind of binary relation.[3] Binary relations are also heavily used in computer science.

A binary relation over sets X and Y is an element of the power set of X × Y. Since the latter set is ordered by inclusion (?), each relation has a place in the lattice of subsets of X × Y.

Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder,[4] Clarence Lewis,[5] and Gunther Schmidt.[6] A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.

In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.

The terms correspondence,[7] dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product X × Y without reference to X and Y, and reserve the term "correspondence" for a binary relation with reference to X and Y.

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https://en.wikipedia.org/wiki/Binary_relation