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| '''Freiling's axiom of symmetry (''AX'')''' is a [[set theory|set-theoretic]] axiom proposed by [[Chris Freiling]]. It is based on intuition of Stuart Davidson
| | If you should accelerate the PC then we have come to the proper place. I will show you, today, five rapid methods to dramatically improve the computer's performance.<br><br>So one day my computer suddenly started being weird. I was so frustrated, considering my files were missing, plus I cannot open the files which I required, and then, suddenly, everything stopped working!<br><br>Registry cleaning is important because the registry may receive crowded and messy when it is left unchecked. False entries send the running system shopping for files plus directories that have long ago been deleted. This takes time plus utilizes precious resources. So, a slowdown inevitably occurs. It is incredibly noticeable when we multitask.<br><br>There are tricks to make a slow computer work efficient and quick. In this article, I may tell we only 3 best strategies or ways to avoid a computer of being slow and instead of which make it quicker and work even much better than before.<br><br>The final step is to make sure which you clean the registry of the computer. The "registry" is a big database that stores significant files, settings & options, plus information. Windows reads the files it demands inside purchase for it to run programs by this database. If the registry gets damaged, infected, or clogged up, then Windows are not able to correctly access the files it needs for it to load up programs. As this happens, problems plus mistakes like the d3d9.dll error occur. To fix this plus avoid future setbacks, you need to download plus run a registry cleaning tool. The very suggested software is the "Frontline [http://bestregistrycleanerfix.com/system-mechanic system mechanic professional]".<br><br>Let's start with all the bad sides initially. The initial cost of the product is surprisingly inexpensive. But, it just comes with one year of updates. After which you need to register to monthly updates. The advantage of which is that best optimizer has enough money plus resources to analysis errors. This technique, you are ensured of safe fixes.<br><br>Another issue with all the cracked variation is the fact that it takes too much time to scan the program and whilst it is very scanning, you can not utilize the computer otherwise. Moreover, there is no technical help to these cracked versions which means should you receive stuck someplace, you can't ask for aid. They even never have any customer service help lines wherein we may call or mail to solve a issues.<br><br>Ally Wood is a expert software reviewer plus has worked inside CNET. Then she is working for her own review software organization to give feedback to the software creator plus has done deep test inside registry cleaner software. After reviewing the many popular registry cleaner, she has written complete review on a review website for we which will be accessed for free. |
| but the mathematics behind it goes back to [[Wacław Sierpiński]].
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| Let ''A'' be the set of functions mapping real numbers in the unit interval [0,1] to countable subsets of the same interval. The axiom ''AX'' states:
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| :For every ''f'' in ''A'', there exist ''x'' and ''y'' such that ''x'' is not in ''f''(''y'') and ''y'' is not in ''f''(''x'').
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| A theorem of Sierpiński says that under the assumptions of ZFC set theory,
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| ''AX'' is equivalent to the negation of the [[continuum hypothesis]] (CH). Sierpiński's theorem answered a question of [[Hugo Steinhaus]] and was proved long before the independence of CH had been established by
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| [[Kurt Gödel]] and [[Paul Cohen (mathematician)|Paul Cohen]].
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| Freiling claims that probabilistic intuition strongly supports this proposition
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| while others disagree. There are several versions of the axiom, some of which
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| are discussed below.
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| ==Freiling's argument==
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| Fix a function ''f'' in ''A''. We will consider a thought experiment that involves throwing two darts at the unit interval. We aren't able to physically determine with infinite accuracy the actual values of the numbers ''x'' and ''y'' that are hit. Likewise, the question of whether "''y'' is in ''f''(''x'')" cannot actually be physically computed. Nevertheless, if ''f'' really ''is'' a function, then this question is a meaningful one and will have a definite "yes" or "no" answer.
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| Now wait until after the first dart, ''x'', is thrown and then assess the chances that the second dart ''y'' will be in ''f''(''x''). Since ''x'' is now fixed, ''f''(''x'') is a fixed countable set and has [[Lebesgue measure]] zero. Therefore this event, with ''x'' fixed, has probability zero. Freiling now makes two generalizations:
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| * Since we can predict with virtual certainty that "''y'' is not in ''f''(''x'')" after the first dart is thrown, and since this prediction is valid no matter what the first dart does, we should be able to make this prediction before the first dart is thrown. This is not to say that we still have a measurable event, rather it is an intuition about the nature of being predictable.
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| * Since "''y'' is not in ''f''(''x'')" is predictably true, by the symmetry of the order in which the darts were thrown (hence the name "axiom of symmetry") we should also be able to predict with virtual certainty that "''x'' is not in ''f''(''y'')".
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| The axiom ''AX'' is now justified based on the principle that what will predictably happen every time this experiment is performed, should at the very least be possible. Hence there should exist two real numbers ''x'', ''y'' such that ''x'' is not in ''f''(''y'') and ''y'' is not in ''f''(''x'').
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| ==Relation to the (Generalised) Continuum Hypothesis==
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| Fix <math>\kappa\,</math> an infinite cardinal (''e.g.'' <math>\aleph_{0}\,</math>). Let <math>\texttt{AX}_{\kappa}.\,</math> be the statement: ''there is no map <math>f:\mathcal{P}(\kappa)\to\mathcal{P}\mathcal{P}(\kappa)\,</math> from sets to sets of size <math>\leq\kappa</math> for which <math>(\forall{x,y\in\mathcal{P}(\kappa)})\,</math> either <math>x\in f(y)\,</math> or <math>y\in f(x)\,</math>.''
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| '''Claim:''' <math>\texttt{ZFC}\vdash 2^{\kappa}=\kappa^{+}\leftrightarrow\neg\texttt{AX}_{\kappa}.\,</math>.
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| '''Proof:'''
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| ''Part I'' (<math>\Rightarrow\,</math>):
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| Suppose <math>2^{\kappa}=\kappa^{+}\,</math>. Then letting <math>\sigma:\kappa^{+}\to\mathcal{P}(\kappa)\,</math> a bijection, we have <math>f:\mathcal{P}(\kappa)\to\mathcal{P}\mathcal{P}(\kappa)\,</math> <math>:\sigma(\alpha)\mapsto \{\sigma(\beta):\beta\preceq\alpha\}\,</math> clearly demonstrates the failure of Freiling's axiom.
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| ''Part II'' (<math>\Leftarrow\,</math>):
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| Suppose that Freiling's axiom fails. Then fix some <math>f\,</math> to verify this fact. Define an order relation on <math>\mathcal{P}(\kappa)\,</math> by <math>A\leq_{f} B</math> iff <math>A\in f(B)</math>. This relation is total and every point has <math>\leq\kappa</math> many predecessors. Define now a strictly increasing chain <math>(A_{\alpha}\in\mathcal{P}(\kappa))_{\alpha<\kappa^{+}}</math> as follows: at each stage choose <math>A_{\alpha}\in\mathcal{P}(\kappa)\setminus\bigcup_{\xi<\alpha}f(A_{\xi})</math>. This process can be carried out since for every ordinal <math>\alpha<\kappa^{+}\,</math>, <math>\bigcup_{\xi<\alpha}f(A_{\xi})\,</math> is a union of <math>\leq\kappa\,</math> many sets of size <math>\leq\kappa\,</math>; thus is of size <math>\leq\kappa<2^{\kappa}\,</math> and so is a strict subset of <math>\mathcal{P}(\kappa)\,</math>. We also have that this sequence is ''cofinal'' in the order defined, ''i.e.'' every member of <math>\mathcal{P}(\kappa)\,</math> is <math>\leq_{f}\,</math> some <math>A_{\alpha}\,</math>. (For otherwise if <math>B\in\mathcal{P}(\kappa)\,</math> is not <math>\leq_{f}\,</math> some <math>A_{\alpha}</math>, then since the order is total <math>(\forall{\alpha<\kappa^{+}})A_{\alpha}\leq_{f} B\,</math>; implying <math>B\,</math> has <math>\geq\kappa^{+}>\kappa\,</math> many predecessors; a contradiction.) Thus we may well-define a map <math>g:\mathcal{P}(\kappa)\to\kappa^{+}\,</math> by <math>B\mapsto\operatorname{min}\{\alpha<\kappa^{+}:B\in f(A_{\alpha})\}</math>.
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| So <math>\mathcal{P}(\kappa)=\bigcup_{\alpha<\kappa^{+}}g^{-1}\{\alpha\}=\bigcup_{\alpha<\kappa^{+}}f(A_{\alpha})\,</math> which is union of <math>\kappa^{+}\,</math> many sets each of size <math>\leq\kappa\,</math>. Hence <math>2^{\kappa}\leq\kappa^{+}\cdot\kappa=\kappa^{+}\,</math> and we are done.
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| {{NumBlk|1=|2=|3=<math>\blacksquare</math> (Claim)|RawN=.}}
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| Note that <math>|[0,1]|=|\mathcal{P}(\aleph_{0})|\,</math> so we can easily rearrange things to obtain that <math>\neg\texttt{CH}\Leftrightarrow\,</math> the above mentioned form of Freiling's axiom.
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| The above can be made more precise: <math>\texttt{ZF}\vdash(\texttt{AC}_{\mathcal{P}(\kappa)}+\neg\texttt{AX}_{\kappa})\leftrightarrow \texttt{CH}_{\kappa}\,</math>. This shows (together the fact that the continuum hypothesis is independent of choice) a precise way in which the (generalised) continuum hypothesis is an extension of the axiom of choice. | |
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| ==Objections to Freiling's argument==
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| Freiling's argument is not widely accepted because of the following two problems with it (which Freiling was well aware of and discussed in his paper).
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| * The naive probabilistic intuition used by Freiling tacitly assumes that there is a well-behaved way to associate a probability to any subset of the reals. But the mathematical formalization of the notion of "[[probability]]" uses the notion of [[Measure (mathematics)|measure]], yet the axiom of choice implies the existence of non-measurable subsets, even of the unit interval. Some examples of this are the [[Banach–Tarski paradox]] and the existence of [[Vitali set]]s.
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| *A minor variation of his argument gives a contradiction with the axiom of choice whether or not one accepts the continuum hypothesis, if one replaces countable additivity of probability by additivity for cardinals less than the continuum. (Freiling used a similar argument to claim that [[Martin's axiom]] is false.) It is not clear why Freiling's intuition should be any less applicable in this instance, if it applies at all. {{harv|Maddy|1988|p=500}} So Freiling's argument seems to be more an argument against the possibility of well ordering the reals than against the continuum hypothesis.
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| ==Connection to graph theory==
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| Using the fact that in ZFC, we have <math>2^{\kappa}=\kappa^{+}\Leftrightarrow\neg\texttt{AX}_{\kappa}\,</math> (see [[Freiling's axiom of symmetry#Relation to the (Generalised) Continuum Hypothesis|above]]), it is not hard to see that the ''failure'' of the axiom of symmetry — and thus the success of <math>2^{\kappa}=\kappa^{+}\,</math> — is equivalent to the following combinatorial principle for graphs:
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| :* The [[complete graph]] on <math>\mathcal{P}(\kappa)\,</math> can be so directed, that every node leads to at most <math>\kappa\,</math>-many nodes.
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| :* In the case of <math>\kappa=\aleph_{0}\,</math>, this translates to: The complete graph on the unit circle can be so directed, that every node leads to at most countably-many nodes.
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| Thus in the context of ZFC, the failure of a Freiling axiom is equivalent to the existence of a specific kind of choice function.
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| ==References==
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| *{{Citation | last1=Freiling | first1=Chris | title=Axioms of symmetry: throwing darts at the real number line | doi=10.2307/2273955 | mr=830085 | year=1986 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=51 | issue=1 | pages=190–200}}
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| * {{cite journal|last=Maddy|first=Penelope|title=Believing the Axioms, I|journal=Journal of Symbolic Logic|volume=53|issue=2|year=1988|pages=481–511|authorlink=Penelope Maddy|doi=10.2307/2274520}}
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| *David Mumford, "The dawning of the age of stochasticity", in ''Mathematics: Frontiers and Perspectives 2000'', American Mathematical Society, 1999, 197–218.
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| *{{Citation | last1=Sierpiński | first1=Wacław | title=Hypothèse du continu | origyear=1934 | publisher=Chelsea Publishing Company, New York, N. Y. | mr=0090558 | year=1956}}
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| *John Simms, "Traditional Cavalieri principles applied to the modern notion of area", ''J. Philosophical Logic'' 18 (1989), 275–314.
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| [[Category:Axioms of set theory]]
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If you should accelerate the PC then we have come to the proper place. I will show you, today, five rapid methods to dramatically improve the computer's performance.
So one day my computer suddenly started being weird. I was so frustrated, considering my files were missing, plus I cannot open the files which I required, and then, suddenly, everything stopped working!
Registry cleaning is important because the registry may receive crowded and messy when it is left unchecked. False entries send the running system shopping for files plus directories that have long ago been deleted. This takes time plus utilizes precious resources. So, a slowdown inevitably occurs. It is incredibly noticeable when we multitask.
There are tricks to make a slow computer work efficient and quick. In this article, I may tell we only 3 best strategies or ways to avoid a computer of being slow and instead of which make it quicker and work even much better than before.
The final step is to make sure which you clean the registry of the computer. The "registry" is a big database that stores significant files, settings & options, plus information. Windows reads the files it demands inside purchase for it to run programs by this database. If the registry gets damaged, infected, or clogged up, then Windows are not able to correctly access the files it needs for it to load up programs. As this happens, problems plus mistakes like the d3d9.dll error occur. To fix this plus avoid future setbacks, you need to download plus run a registry cleaning tool. The very suggested software is the "Frontline system mechanic professional".
Let's start with all the bad sides initially. The initial cost of the product is surprisingly inexpensive. But, it just comes with one year of updates. After which you need to register to monthly updates. The advantage of which is that best optimizer has enough money plus resources to analysis errors. This technique, you are ensured of safe fixes.
Another issue with all the cracked variation is the fact that it takes too much time to scan the program and whilst it is very scanning, you can not utilize the computer otherwise. Moreover, there is no technical help to these cracked versions which means should you receive stuck someplace, you can't ask for aid. They even never have any customer service help lines wherein we may call or mail to solve a issues.
Ally Wood is a expert software reviewer plus has worked inside CNET. Then she is working for her own review software organization to give feedback to the software creator plus has done deep test inside registry cleaner software. After reviewing the many popular registry cleaner, she has written complete review on a review website for we which will be accessed for free.