Sellmeier equation

Freiling's axiom of symmetry (AX) is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidson but the mathematics behind it goes back to Wacław Sierpiński.

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:

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).

A theorem of Sierpiński says that under the assumptions of ZFC set theory, 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 Kurt Gödel and Paul Cohen.

Freiling claims that probabilistic intuition strongly supports this proposition while others disagree. There are several versions of the axiom, some of which are discussed below.

Freiling's argument

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.

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:

• 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.

• 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)".

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).

Relation to the (Generalised) Continuum Hypothesis

Fix $\kappa$ an infinite cardinal (e.g. ${\mathrm{\aleph }}_0$). Let ${\mathtt{\text{<mrow data-mjx-texclass="ORD">AX</mrow>}}}_{\kappa }.$ be the statement: there is no map $f:\mathcal{𝒫}(\kappa )\to \mathcal{𝒫}\mathcal{𝒫}(\kappa )$ from sets to sets of size $\le \kappa$ for which $(\mathrm{\forall }x,y\in \mathcal{𝒫}(\kappa ))$ either $x\in f(y)$ or $y\in f(x)$.

Claim: $\mathtt{\text{<mrow data-mjx-texclass="ORD">ZFC</mrow>}}⊢2^{\kappa }={\kappa }^{+}\leftrightarrow \mathrm{\neg }{\mathtt{\text{<mrow data-mjx-texclass="ORD">AX</mrow>}}}_{\kappa }.$.

Proof: Part I ($\Rightarrow$):

Suppose $2^{\kappa }={\kappa }^{+}$. Then letting $\sigma :{\kappa }^{+}\to \mathcal{𝒫}(\kappa )$ a bijection, we have $f:\mathcal{𝒫}(\kappa )\to \mathcal{𝒫}\mathcal{𝒫}(\kappa )$ $:\sigma (\alpha )\mapsto \{\sigma (\beta ):\beta ⪯\alpha \}$ clearly demonstrates the failure of Freiling's axiom.

Part II ($\Leftarrow$):

Suppose that Freiling's axiom fails. Then fix some $f$ to verify this fact. Define an order relation on $\mathcal{𝒫}(\kappa )$ by $A{\le }_{f}B$ iff $A\in f(B)$. This relation is total and every point has $\le \kappa$ many predecessors. Define now a strictly increasing chain $(A_{\alpha }\in \mathcal{𝒫}(\kappa ){)}_{\alpha <{\kappa }^{+}}$ as follows: at each stage choose $A_{\alpha }\in \mathcal{𝒫}(\kappa )\setminus \bigcup _{\xi <\alpha }f(A_{\xi })$. This process can be carried out since for every ordinal $\alpha <{\kappa }^{+}$, $\bigcup _{\xi <\alpha }f(A_{\xi })$ is a union of $\le \kappa$ many sets of size $\le \kappa$; thus is of size $\le \kappa <2^{\kappa }$ and so is a strict subset of $\mathcal{𝒫}(\kappa )$. We also have that this sequence is cofinal in the order defined, i.e. every member of $\mathcal{𝒫}(\kappa )$ is ${\le }_f$ some $A_{\alpha }$. (For otherwise if $B\in \mathcal{𝒫}(\kappa )$ is not ${\le }_f$ some $A_{\alpha }$, then since the order is total $(\mathrm{\forall }\alpha <{\kappa }^{+})A_{\alpha }{\le }_{f}B$; implying $B$ has $\ge {\kappa }^{+}>\kappa$ many predecessors; a contradiction.) Thus we may well-define a map $g:\mathcal{𝒫}(\kappa )\to {\kappa }^{+}$ by $B\mapsto \min \{\alpha <{\kappa }^{+}:B\in f(A_{\alpha })\}$. So $\mathcal{𝒫}(\kappa )=\bigcup _{\alpha <{\kappa }^{+}}{g}^{-1}\{\alpha \}=\bigcup _{\alpha <{\kappa }^{+}}f(A_{\alpha })$ which is union of ${\kappa }^{+}$ many sets each of size $\le \kappa$. Hence $2^{\kappa }\le {\kappa }^{+}\cdot \kappa ={\kappa }^{+}$ and we are done. Template:NumBlk

Note that $|[0,1]|=|\mathcal{𝒫}({\mathrm{\aleph }}_0)|$ so we can easily rearrange things to obtain that $\mathrm{\neg }\mathtt{\text{<mrow data-mjx-texclass="ORD">CH</mrow>}}\iff$ the above mentioned form of Freiling's axiom.

The above can be made more precise: $\mathtt{\text{<mrow data-mjx-texclass="ORD">ZF</mrow>}}⊢({\mathtt{\text{<mrow data-mjx-texclass="ORD">AC</mrow>}}}_{\mathcal{𝒫}(\kappa )}+\mathrm{\neg }{\mathtt{\text{<mrow data-mjx-texclass="ORD">AX</mrow>}}}_{\kappa })\leftrightarrow {\mathtt{\text{<mrow data-mjx-texclass="ORD">CH</mrow>}}}_{\kappa }$. 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.

Objections to Freiling's argument

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).

• 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, 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 sets.
• 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. Template:Harv So Freiling's argument seems to be more an argument against the possibility of well ordering the reals than against the continuum hypothesis.

Connection to graph theory

Using the fact that in ZFC, we have $2^{\kappa }={\kappa }^{+}\iff \mathrm{\neg }{\mathtt{\text{<mrow data-mjx-texclass="ORD">AX</mrow>}}}_{\kappa }$ (see above), it is not hard to see that the failure of the axiom of symmetry — and thus the success of $2^{\kappa }={\kappa }^{+}$ — is equivalent to the following combinatorial principle for graphs:

• The complete graph on $\mathcal{𝒫}(\kappa )$ can be so directed, that every node leads to at most $\kappa$-many nodes.
• In the case of $\kappa ={\mathrm{\aleph }}_0$, this translates to: The complete graph on the unit circle can be so directed, that every node leads to at most countably-many nodes.

Thus in the context of ZFC, the failure of a Freiling axiom is equivalent to the existence of a specific kind of choice function.

References

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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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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

• John Simms, "Traditional Cavalieri principles applied to the modern notion of area", J. Philosophical Logic 18 (1989), 275–314.