# Home Order, philosophy: order is the division of a subject area by distinctions or the highlighting of certain differences as opposed to other differences. The resulting order can be one-dimensional or multi-dimensional, i.e. linear or spatial. Examples are family trees, lexicons, lists, alphabets. It may be that only an order makes certain characteristics visible, e.g. contour lines. Ordering spaces may be more than three-dimensional, e.g. in the attribution of temperatures to color-determined objects. See also conceptual space, hierarchies, distinctness, indistinguishability, stratification, identification, individuation, specification.

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Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

Author Item Summary Meta data
Thiel I 201
Order/Mathematics/Thiel:
Def Well-ordered: if an ordered set Mp is such that not only it itself, but also each of its non-empty subsets has such an element, in the sense of the order first element, then we call M< a well-ordered set.
I 201/202
Well-ordered sets are special ordered sets, therefore each pairterm represents an order type for a well-ordered set and the order types can now be shown to be comparable with each other. In this sense, the order types of well-ordered sets are more "number-like" than other order types.
We call them
Def Ordinal Numbers. The order type of a finite set (which is well-ordered in any arrangement) coincides with its ordinal number and beyond that with its thickness.
I 201f
Def well-ordered: is a set, if every non-empty subset has a first element - i.e. every pairterm is also an order type - then all order types are comparable.
Addition, multiplication potentiation especially: Example {1, 1, 2, 2..} shall be mapped to the naturally ordered set of basic numbers...I 202 Example {1,3,5...;2,4,6...} non-commutative.
Terminology: ordinal number ω.
In the case of ordinal numbers we can thus in a very specific sense go beyond the ordinal number ω of the naturally ordered set of basic numbers:
The elements of a set of the power Ao of the basic numbers can still be ordered in various ways and thus lead to quite different transfinite ordinal numbers
I 203
and quite different well-orders of these sets lead also in the indicated sense to "larger" ordinal numbers than .
But one should not jump to conclusions about a deeper penetration into the realm of the infinite, because an ordered set with the ordinal number ω exp ω does not have the power of Ao exp Ao (which according to classical view would be the power of the continuum), but is still countable, i.e. of the same power Ao as an ordered set with the ordinal number ω.
Without the condition that every quantity can be well ordered, which has not been substantiated up to now anywhere, one cannot reach higher powers.
I 203
ω exp ω is still countable. Against: Power of the Continuum: Ao exp Ao
ConstructivismVsCantor: Objection to the introduction of absolute transfinite numbers: arises from the definition of uniformity and similarity. They take place with recourse to illustration.
According to the constructivist view, each representation must be represented as a function by a function term.
However, this must refer to a fixed inventory of permitted mathematical means of expression. An illustration is then expressable or not.
Example: The uniformity of two sets can be expressable in a formal system F1 (thus "exist") in another F2 however not.
For a Platonist, of course, this is an untenable situation. He will say that the system F2 is simply too "weak in expression".
The system would have to be extended. But according to the constructivists this is not possible: forbidden, because the means of expression necessary for their representation (or set-theoretical axioms, which would first secure the "existence" of the representation in question), would lead to a contradiction with the other means of expression or axioms.
There is no known possibility to introduce transfinite cardinal numbers (and in axiomatic systems also transfinite ordinal numbers) as absolute milestones in infinity in a harmless way.

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Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.

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Chr. Thiel