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The recursive nature of an expanded and compressed infinity...viewing two well-known infinities from inside and out. Version 1 revision 3 - 11th of November, 2023 This simplifies and supersedes a previous document named "An alternative way to visualize infinity". by Steve Sybesma ====================================================================================================================== Introduction to concepts: Infinity is normally visualized by us in this conventional/natural way: a. by the fact we 'live outside' the infinity of decimal values that exist between zero and '1' and we CAN see both 'ends'. b. by the fact we 'live inside' the infinity of integers that exists between zero and 'infinity' and we CANNOT see both 'ends'; because we are stuck seeing it this way, most experts claim this infinity of integers cannot possibly have an end, even though the other infinity clearly does. But notice how these infinities can be turned inside out. In doing so, some very interesting properties come to light. Try imagining yourself doing this: a. consider you are starting outside the infinity of decimal values and place yourself INSIDE of it (let's start at zero) so that you can see individual 'points' which become just as countable as integers but go on with no end...the same as the infinity of integers; you can name the points after zero "infinitesimal 1", infinitesimal 2", infinitesimal 3", etc... b. consider you are starting inside the infinity of integers and place yourself OUTSIDE of it so that only the 'ends' of it are visible (yes, infinity looks like a line segment if you're doing this exercise properly) and no points between are visible. If you do this thought experiment properly, the first looks virtually like the normal number line and the second looks virtually like a line segment of a number line starting with zero and ending with a unit or whole called 'infinity', and which can be seen as a type of '1'. This new viewpoint changes infinity from becoming a mere concept into a type of number (a whole or a unit). Begin this explanation of a new way to see both infinities and show there is a recursive relationship: 1.Take what are considered two different and well-known infinities: a. the one that exists between zero and one consisting of uncountable infinitesimal points b. the one that exists from zero onward consisting of uncountable integers With "a", infinity is enclosed by limits at both ends, yet somehow through Cantor's 'parlor trick' it's said to be a larger infinity than "b". Cantor's proof attempted to show that "a" is larger but I say his logic is flawed because it did much more to obfuscate reality than showing it and his was not a head-on approach to reality but a misleading sidestep. It is that, because infinity if not carefully dealt with defies normal arithmetic and easily leads to misleading conclusions. I intend to disprove Cantor and show both infinities memberships are absolutely equal through a much 'purer' method that treats both types of infinities in the same manner by declaring they are two opposing views of the same infinity. 2.Basic properties of the two infinities: a. consider 1a shows an infinity that has two 'ends', zero and one and that the smallest value above zero can never be determined using the normal convention; this is what I consider a 'compressed' view, typical of this form of infinity. b. consider 1b shows an infinity that has only one 'end' and that the largest member below infinity can never be determined using the normal convention; this is what I consider an 'expanded' view, typical of this form of infinity. Now, what happens if you allow the thought experiment to expand the first and compress the second, thereby flipping their views? 3.The resulting observation: a. When 2a is viewed from the 'inside' (expanded view), you can distinguish members you could not before. What was '1' now resembles another type of infinity. You can see the first member after zero (before '1'); logically it has to be a number ending with '1' with infinite preceding zeros after the decimal; this by the way is also how the normal integer number line looks, just remove the decimal point. Adjacent members can be plotted; before adjacent members could not be plotted. The number line looks like the integers starting with zero (just remove decimals and leading zeroes). b. When 2b is viewed from the 'outside' (compressed view), you cannot distinguish members you could before. What was infinity resembles another type of '1'. You cannot see the first member after zero (except infinity); all numbers you can name are virtually at the same location as zero. Adjacent members cannot be plotted; before adjacent members could be plotted. The number line looks like the integers zero and '1', any 'fraction' of infinity between behaves like and must be written out like a decimal number between zero and '1'; I suggest creating a new character called a 'super-decimal' to represent such values which looks like a decimal point except that it's at the top of the character block instead of the bottom; its use is to represent a fraction of infinity the same way a decimal is used to represent a fraction of an integer. We say infinity is not a number...in a sense (only because of how we view it) that's absolutely true, but it can look like an integer if the view of the number line is compressed. That integer would be '1'. More accurately using the new character I suggested above, it would look like 1.0 with the decimal moved to the top of the character block (the super-decimal). This new convention would make possible calculations involving a whole or a part of infinity. Notice the two infinities are recursive (the infinitesimally smaller infinity is nested within the larger infinity and is nothing more or less than an enclosed miniaturization). The views of each are essentially a perfect opposite of the other. When both are flipped, the smaller looks virtually like the larger and the larger looks virtually like the smaller. Any differences are cosmetic (decimal vs. no decimal). When you observed 3a and 3b after performing the thought experiment properly, you will see clearly HOW it is possible that the two infinities have the same exact set membership (the size of the sets are absolutely perfectly equal). What is true about both sets yet does not affect my explanation showing how the members are absolutely equal: a.Members are not truly name-able numbers if they must be written out using INFINITE digits; this is true even of decimal numbers in the sense the values reaching the lowest significant digit cannot be reached, and hence named; yes that means irrational and transcendental numbers are not truly name-able numbers, but that does not mean they are not USABLE numbers because we can write out the most significant digits; the compressed view we normally use for decimal numbers allows for that and would not work if the expanded view were used because you would be forced to start writing out the number with the lowest significant digit and could never reach the highest significant digit b.members are name-able numbers if they can be written out using finite digits (This is true regardless if you're talking about membership in either set; infinite digits define un-nameability) Important note on aspect I noticed which can easily be dealt with if you consider the two different views of the number line: The decimal numbers begin naming their 'places' with the most significant digit, making the least significant possible digit un-nameable The integers begin naming their 'places' with the least significant digit, making the most significant possible digit un-nameable (When the views are flipped, it becomes possible to do what was not possible before, and vice versa; the plotting of infinitesimals after zero becomes possible and starts with "1 infinitesimal, 2 infinitesimal, etc... which are now all VISIBLE since the number line was EXPANDED; it now becomes impossible to plot integers because they are INVISIBLE since the number line was COMPRESSED, yet you still know because of our BIAS/TRAINING toward the normal convention that the now invisible integers after zero start with '1, 2, etc...all these integer 'points' exist the same way infinitesimals exist but it is the EXPANDED or COMPRESSED view that determine visibility. It is only our bias/training toward conventional reality that causes whatever is not visible to appear not to exist.) Exploring deeper into why the sets are absolutely, perfectly equal: The expansion of the members between zero and '1' makes it possible to do a 1:1 match to the members between zero and infinity because the memberships both resemble integers starting from zero toward infinity ('1' being a 'type' of infinity) The compression of the members between zero and infinity makes it possible to do a 1:1 match of the members between zero and '1' because the memberships both resemble decimal values between zero and '1' (infinity being a 'type' of '1' Note that there is a 'type' similarity of '1' to infinity as they both represent a 'whole' of something; '1' represents a 'whole' of all the possible decimal values leading up to itself; infinity represents a 'whole' of all the possible integer values leading up to itself; in that sense they are two ways of describing the same concept

Conjecture - All paradoxes reduce to Banach-Tarski. In set-theoretic terms, it is all boils down to the objectification of an empty set. To paraphrase Hamlet, "To objectify or not to objectify, that is the question." All paradoxes are contradictions that constitute an attempt to reconcile a gestalt - an accounting of an object and the space that it occupies. I.e., A sphere and the space that the sphere occupies. Does the sphere actually "occupy" an equivalent space or does the sphere move in and "push aside" an equivalent amount of space? Does this problem require an assembly of two spheres - one composed of matter and the other of space? There are always two perspectives in the attempt to reconcile the coexsistence of object and space. Positing a set as an element of itself is one viewpoint, Russell's considers the same from another viewpoint in an "anti-set" configuration. Consider Hilbert's Hotel, an infinite set of two-element sbsets comprised of a gestalt pairing of (occupant, hotel room. The problem begins with the introduction of a one element subset from outside the Hilbert set and the paradox is ignited. Hilberts proposition of moving the occupant from room to room does not resolve the paradox, it merely kicks the can down the road. Compare that with the problem of the unification of mathematical physics in an attempt to reconcile spacetime with mass/energy..

Perhaps Pure Mathematics, itself, is the source of all paradox. Every number and every element is infinitely duplicable by convention. There is an infinite supply of each number - I can take any number and multiply it by itself an infinite number of times. Inexaustability is built into the program from the start. That is what makes mathematics universally functional. In the real world, I cannot pull an infinite number of rabbits out of a hat.

Considering the notion of a "set of all sets" as a member of itself: Doesn't this really amount to a Banach-Tarski doubling manipulation? In effect, by the inclusion it would seem you are adding an additional set of all sets to the set of all sets. It is like a mathematical set of siamese twins. And where does this additional set come from?

In the proposal of Hilbert's Hotel you have a set of infinite rooms, each of which has an occupant. But then, from somewhere outside this described environment, a new potential customer is introduced and the continuous room switching commences. This outsider here is key. Please note that the new customer is introduced from outside the infinity and as the room switching takes place there must always be a customer outside a room. Hence, you will always have an occupant that is not an occupant in the mix. Something very much like Russel's paradox seems to have been introduced by setting the hotel occupancies in motion. The attempted manipulation of the infinte by an outsider generally always introduces paradox, when examined closely. Perhaps this is where Schroedinger and Bohr found the notion of observer-created reality to be remedial to wave-function collapse. The potential and the actualized are always in flux.

There is a barber who shaves today all and only those people who did not shave themselves yesterday. Suppose that the barber shaved himself yesterday. Then the barber does not shave himself today. Suppose that the barber did not shave himself yesterday, then the barber shaves himself today. So the barber shaves himself today if and only if he didn't shave himself yesterday. Contradiction gone with some relative tense differentiation between the clauses. Ditto for Russell's set: There is a set that includes now all and only those sets that did not include themselves previously. Suppose that the set included itself previously. Then the set does not include itself now. Suppose that the set did not include itself previously, etc etc. Think of set inclusion as a mathematical operation like arithmetical addition, performed at a particular time and possibly one of a sequence of such operations performed at different times. When you write AεB you are not making a true or false statement, any more than when you write 2+3, but performing an operation of set inclusion, Similarly for ~(AεB), which can be seen as an act of set exclusion. Two arithmetical operations that fail to make a relative tense differentiation, like 2+3*4 can easily be resolved with the use of brackets, or some other way of making explicit which comes first such as add 2 and 3 then multiply by 4, or multiply 3 by 4 then add 2. The Russell "paradox" is therefore just an order of operations ambiguity.