Base case: If the ordered set only has one element, it has the exact same structure as any other ordered set of size one.
Inductive step: Assume that every lineär order of size n has the same structure.
Look at any two lineär orders of size n+1. Because these lineärly ordered sets are finite, they have a minimal element.
This minimal element of a lineär order is unique, by contradiction. If you assume there are two minimal elements, they can be compared, making one strictly greater than the other, which makes it not minimal.
So, the two lineär orders of size n+1 have the same structure of that minimum element and how it relates to the other n elements. So, we have shown the only way they can have different structure is in the n non-minimal elements.
Because the set of non-minimum elements is of size n, by assumption, they have the same structure. So, the whole sets of size n+1 have the same structure.
Combining: If we call the property being true of number i “S(i),” then we know that S(0), the base case, and S(n)->S(n+1), which makes S(i) propagate to all positive integers.
That is, we proved that for all finite sizes, there is only one unique (up to isomorphism!) lineär order.
BUT: It is not true for infinite sets. The positive integers have the same size as the set of all integers. However, with the normal order, one set has a minimum, and the other doesn’t have a lowest element, so they have different order types.
King Solomon / working on your individual PRIORITIES . If you have been keeping up with the past few #dailyoracle readings you will know that there were times this week that source was asking you to take time to rest and to honor your feelings. Also messages were received about asking for what you want and starting the reputation you’d like to uphold. Today traditionally is a day of rest but as King Solomon arrives he is letting us know this week will be busy with tasks of organizing and manifesting. There is a sense of relaxed delegations as you ask SPIRIT and your earthly friends to help bring into being some very important goals that are in alignment with your higher self. What is “the work week”? Try to rewire your concept of vacations, lazy sundays and manic mondays! Its all a thing of the past as we open ourselves to new ways of understanding our own personal rhythm. If there are days where you are tired, try to inhabit a relaxed state of mind, filled with moment of deep breath, bathroom breaks to recuperate and center as well as gratitude for your nervous system and heart and all the work they do for you! When you begin to listen to your mind, body and soul you will be much more productive! ITs a journey! take your time, and continue to make this a practice that lasts! The benefit is being in alignment with your higher self and with the betterment of this planet and all of its inhabitants! All by working on getting to know what is right for you!! <3 King Solomon is not a frantic and fried king. He works with the order of the universe and his inner guidance! He rules with joy, confidence in spirit and psychic awareness! Take two finger bring them up to the space between your brows and chant OM three times are you gently tap, tap, tap and awaken the #third eye! I love you! If you are interested in finding out a personal message for your life journey contact me at Advice.firstname.lastname@example.org Triplegoddesstarotandhealing.com #kingsolomon #health #wealth #prosperity #thirdeye #om #higherpath #ascendedmasters #dailytarot #metaphysical #cosmic #order #oraclecards #doreenvirtue #priorities #individual Crystal in this photo : #unakite
This is the awe-inspiring universe of magic: There are no atoms, only waves and motions all around. Here, you discard all belief in barriers to understanding. You put aside understanding itself. This universe cannot be seen, cannot be heard, cannot be detected in any way by fixed perceptions. It is the ultimate void where no preordained screens occur upon which forms may be projected. You have only one awareness here—the screen of the magi: Imagination! Here, you learn what it is to be human. You are a creator of order, of beautiful shapes and system, an organizer of chaos.
The Patrician didn’t believe in unnecessary cruelty.* He did not believe in pointless revenge. But he was a great believer in the need for things to be sorted out.
*While being bang alongside the idea of necessary cruelty, of course.
These fiercely stylish folks come from the Catalogo degli Ordini Equestri e Militari, written and engraved by Filippo Bonanni and published in 1711. This delightful volume (shown in the first image next to an identical one) is filled to bursting with detailed engravings such as the ones above, as well as descriptions of the history of the orders and prominent members. Of particular note in this selection of engravings are the Peruvian and Floridian individuals in the 7th and 8th images- these are relatively early engravings of natives in a non-objectified context.
Bonanni, an Italian Jesuit scholar perhaps most famous for his engraving of a flea under a microscope (the first time a specimen had been engraved in such detail), honed his engraving skills in the field of conchology and published the first book solely on the topic of seashells.
And, in the spirit of a recent meme, tag yourself! I’m Ordo Foeminarum dictus a Cruce.
(From the collection of the Royal Scottish Academy of Art and Architecture in Edinburgh, The Dean Gallery)
This order is not to be understood solely in terms of a regular arrangement of objects (e.g., in rows) or as a regular arrangement of events (e.g. in a series). Rather, a total order is contained in some implicit sense, in each region of space and time. Now the word ‘implicit’ is based on the verb 'to implicate’. This means 'to fold inward’ (as multiplication means 'folding many times’). So we may be led to explore the notion that in some sense each region contains a total structure 'enfolded’ within it.