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What is the geometrical definition of point and what is the definition of completeness in the power set of cardinality higher than real number cardinality

moh
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Modern geometry does not define the word "point": it is an undefined term. (See here or in the high school textbook Geometry by Burger et al., page 6.) Other terms are often defined in terms of it. It is a basic building block of geometry. If undefined, geometry can be applied to areas other than what we normally think. The concept is "defined" by the axioms that points must satisfy, often in relation to other items.

There are many kinds of completeness, but it usually relates to a totally ordered set. You did not mention total order, so your second question cannot be answered. The ordinal numbers do have a total order, but they are not complete since each ordinal number has an immediate successor.

There is also algebraic completeness, but that also is not relevant here.

Rory Daulton
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  • First thanks a lot and I mean completness that have no gaps like real line but what is the in your opinion the relation between my completness meaning that i ask and the other definition and is there a book that discuss the problem of the point in modern geometry – moh Nov 12 '14 at 01:19
  • I also feel the point is one of the blocks and you confirm that feeling but are there researches on the relation of the point please your answer – moh Nov 12 '14 at 01:29
  • "Gaps" make sense only in a total order. The ordinal numbers of any cardinality do have gaps, for example between any two consecutive ordinal numbers, so they are not complete. Do you mean something else? As for points, just about any modern geometry textbook says that points are undefined. I edited my answer to give an example. As for "researches on the relation of the point" I do not know what you mean. Please explain. – Rory Daulton Nov 12 '14 at 01:52
  • Iwant to ask about like natural numbers have gaps and the real number doesnot have as we know but in higher cardinality like the power set of real number and the power set of the power set of real numbers and so on how the term gap in these set and can we making like real line in these power sets with some of harmony in definition of line gap like the line in aleph and 2 power aleph – moh Nov 12 '14 at 02:02
  • I mean is there a definition that is in harmony with the terms of gaps and line in power sets of real numbers and so on – moh Nov 12 '14 at 02:22
  • A power set does not have a natural total order, so your talk about gaps makes no sense. There are several ways to well-order a power set, given a well-order on the base set, but none of these are complete, since a well-order has a gap after every member. Every other order I can think of on a power set is not complete. Note that the real numbers are not a power set--they just have the same cardinality as the power set of the rationals. – Rory Daulton Nov 12 '14 at 10:57
  • You are welcome, but the usual way to give thanks at this site is to vote an answer up or accept the answer. If you continue at this site you should get in the habit of doing those. – Rory Daulton Nov 12 '14 at 20:19
  • for voting I am anew member and I must have some reputations to vote and again thanks a lot but I will ask also some questions differs from that and I hope to answer again Mr.Rory but can I take your opinion about my question of the harmony of definition of gaps to be extended to power sets with more cardinality than real number cardinal I am sorry to begin with some un definied question but I mean the harmony is it good question or not for an amateur? – moh Nov 12 '14 at 23:04