There has been much hype, debate, and confusion over the Affordable Care Act lately. Let’s try a different approach: let’s look at the facts and see on whom costs or benefits are imposed as a consequence of the upholding of this legislation. A key list of the provisions in the Act can be found at http://www.healthcare.gov/law/timeline/full.html. It possesses a series of provisions to take effect beginning in 2010 and extending to 2015.
2010 (consumer protections): It sets up a website where consumers can view various insurance programs and weigh options. This will no doubt increase competition in the insurance market. It will also impose restrictions on how health insurance companies charge: children under 19 cannot be denied coverage because of a pre-existing condition, those who acquire insurance cannot be dropped later due to a clerical error on a customer’s application (this tended to be a strategy used by some insurance companies to drop coverage for customers who acquired a sickness over the course of their coverage), it sets up an external review process that enables consumers to appeal decisions made by their healthcare providers regarding the determination of their pricing, small businesses are eligible for tax credits to help pay for their employees’ coverage, new plans must include preventive care like mammograms and colonoscopies, young adults without a plan (or offered plan) may stay on parents’ plan until they turn 26, and many other features in 2010.
2011-12: Incentives for improving the quality and lowering the cost of medical care, special features for seniors. For insurance companies who sell to large employers, at least 85% of their revenue from sold policies must go directly to healthcare providing. And for insurance companies who sell to small businesses and individuals, at least 80% of their revenue from policies must go directly to healthcare. This provision is aimed at reducing the consolidation of profits into the form of administrative costs and executive bonuses.
2013 (Medicaid improvements): bundling costs, subsidizing state Medicaid while sustaining pay for primary care physicians.
2014: insurance companies with individual and small group programs are no longer allowed to charge different rates based on gender or health, small businesses and those not covered by an employer will be able to buy insurance in an Exchange program (a competitive market for individual and small groups) (members of Congress will purchase it from here as well), more tax credits to small businesses to help cover employees, those at or below 133% of the poverty line will be eligible for Medicaid, most Americans who can afford healthcare will be required to purchase a plan or pay a fee to help cover costs of others, if healthcare is not available to a person then they will be eligible to apply for an exemption, workers who can’t afford their employer’s program will be able to keep their insurance funds that would have otherwise been taken out of their paycheck.
It’s clear that this Act has the intention to benefit anyone who has or wants health insurance (which is essentially everyone). Who could possibly hurt from this policy? Well, obviously the insurance companies themselves might hurt–as they have less freedom to charge what they want (although at the same time, the individual mandate provision that requires those who can afford it to buy a health insurance policy for his/herself most certainly will increase demand for insurance companies). And businesses that have to cover their employees now face a little more regulation. This is essentially from where the criticism on the right comes. But let’s be honest, large employers and insurance companies who sell to them are likely to not be dinged by these policies. Their profits are sufficient. Small businesses and small insurance providers would be the only ones facing tough changes. And that’s precisely the purpose of things like the tax credits, Exchange program, and the extra 5% of kept revenue (I’m not sure how beneficial the extra 5% would be).
So still, with all of this, how could there still be such opposition to the Affordable Care Act? Since the Citizens United ruling, it’s obvious that big companies have more influence in political elections now relative to, say, everyone else. And where have big companies decided to utilize their “free speech”? They sent money to super PACs of course. And is there anything noteworthy about the choice of super PACs to which they donated? Well consider the 2011 findings by the nonpartisan Center for Responsive Politics :
The centerpiece of how a business becomes very successful (i.e. profitable) is based on the fact that they care very much about profits. It may therefore be no surprise why a party that receives far more funding from big businesses (including insurance companies) and wealthy individuals than another party tends to advocate policies that benefit those contributors. The same can be said of liberals and unions (although one key distinction to be made is that unions contribute far less to liberal candidates than businesses do to conservative ones) . And if big businesses now have to purchase policies from insurance companies that have been slightly constrained by new legislation, it’s no wonder why they would oppose the legislation as it could lead to higher policy prices.
An alternative to the Affordable Care Act (which Obama originally advocated), was a single-payer plan (aka universal healthcare), in which each person paid a tax to cover insurance for everyone–provided by the government. No doubt this would put the health insurance industry out of business. Correspondingly, it’s also no surprise why a party whose campaign was heavily supported by insurance companies also advocated against a potential government competitor:
Top (Insurance) Contributors, 2011-2012
|New York Life Insurance||$1,648,546|
|Blue Cross/Blue Shield||$1,372,945|
|Indep Insurance Agents & Brokers/America||$877,250|
|Massachusetts Mutual Life Insurance||$757,178|
|Natl Assn/Insurance & Financial Advisors||$727,000|
|American Financial Group||$563,950|
|Assn for Advanced Life Underwriting||$548,500|
|National Assn of Health Underwriters||$492,850|
|Property Casualty Insurers Assn/America||$488,500|
|Council of Insurance Agents & Brokers||$480,899|
|Zurich Financial Services||$374,815|
|State Farm Insurance||$333,835|
As the Center for Responsive Politics articulately explained:
Insurance companies staunchly oppose the idea of a government-provided health insurance option, which President Barack Obama and most congressional Democrats support. These businesses fear that implementing a “public option” will eventually lead to “single-payer” health care, which they say would mean the collapse of their industry. Insurers believe that even if they survive the presence of a government competitor in the market, their profits will decline sharply, as the federal government will be able to negotiate for lower premiums and drug costs. 
Unless otherwise specified, we will assume ZF axioms. Recall the following ZF axiom:
Axiom of Infinity:
We will call the set of natural numbers. That is,
We can then define and iterate as before, whence by applying the axiom of infinity again we can obtain and so on. All such sets generated by this process are called ordinals. This in turn gives us the canonical linear ordering of the ordinals where iff
Definition 1. An ordinal is a successor ordinal iff A limit ordinal is an ordinal which is not a successor ordinal.
Proposition 2. (Transfinite Induction) Let denote the class of all ordinals (in accordance to VBG notion of class) and be a class. If
- if is a limit ordinal and for all then
Proof. Since is a linear ordering on the ordinals, let be the least ordinal such that But then which is a contradiction. Hence
We can index ordinals with ordinals to generate the notion of a sequence of ordinals. We define an nondecreasing (nonincreasing) sequence of ordinals an ordered set of ordinals where iff
Definition 3. Let be an nondecreasing sequence of ordinals and be a limit ordinal and Then we define the limit of the sequence as
A dual definition can be defined for nonincreasing sequences, in which case the limits can be respectively distinguished as left and right limits. A sequence is continuous if for every limit ordinal in the indexing subclass we have
An example of a sequence which is not continuous may be one of the form where are both limit ordinals. So in this case
(since the sup is actually a max in this case).
Definition 4. (Ordinal Arithmetic) We define
It follows that addition and multiplication are both associative, but not commutative. In particular one can see that
Definition 5. For a set we define its cardinality, denoted as the unique ordinal with which the set has a bijection. The corresponding subclass of ordinals is called the class of cardinals.
Proposition 6. If then
Proof. For every define
Hence the mapping is a bijection between and
Hence in this context, Cantor’s theorem immediately follows:
Proposition 7. Let and Then
Proof. We have
Also if and are bijections, then we can define by and this is easily seen to be a bijection.
And if then we can define which is also seen to be a bijection.
It thus follows that addition and multiplication of cardinals are commutative (that is, ordinal operations are commutative on this subclass).
Above we said and But
for any finite ordinal with Knuth notation
having raisings of Consider the following convention for defining countable infinite ordinals:
They are all called countable infinite ordinals since for any one of them To clarify when we are talking about cardinal numbers versus ordinal numbers, we will use aleph notation: From Cantor’s theorem above, we know that if then That is, the ordinal corresponding to must be greater than all of the countable ordinals above, otherwise its cardinality would be This necessitates the notion of uncountable ordinals and corresponding uncountable cardinals. We use subscripts to characterize these: etc.
Definition 8. An infinite cardinal is a successor cardinal iff is a successor ordinal, and it is a limit cardinal iff is a limit ordinal.
Definition 9. Let be a limit ordinal. An increasing –sequence with a limit ordinal is cofinal in if And if is an ordinal, then we define its cofinality as
It is easy to verify that iff is a successor ordinal. Also and for any finite ordinal
Proof. Let Then (in particular it is the smallest such ). Now if then certainly Now since is cofinal in a subsequence of indices is cofinal in (where cofinality can be chosen to be ). So
But then is cofinal in That is,
Definition 11. An ordinal is regular if It is singular if it is not regular.
Corollary 12. If is a limit ordinal, then is a regular cardinal.
Theorem 13. If is an infinite cardinal, then
 Jech, Thomas. Set Theory. 3rd Edition. Springer Monographs in Mathematics. Springer-Verlag. 2000.
As it is no surprise about the education in the US, science in particular has also been suffering. Last week in the April meeting of the American Physics Society, a group of physicists made precisely this claim. The primary data used in the basis of their argument was the decline in degrees in science. This data is certainly consistent with the argument that the US is falling behind in science. There is no debate that good performance in science predicates the choice of a degree in science. That is to say, the decline in degrees in science is mostly a consequence of poor performance in science earlier–rather than the cause of it. So the natural question is: why are kids less interested in science?
The everyday life of a kid is governed by three structures: school, parents, and other media they encounter (including television, internet, and social structure). It follows that one should be able to attribute the decline in scientific interest to one or more of these structures. Suppose we assume two things: (1) that there is a problem at some generation with respect to science advocacy in some of these components, and (2) that the extent to which kids assimilate to the scientific understanding of their mentors (i.e. parents and teachers) is strictly less than complete (that is, in an abstract isolated teacher-student situation, the student can only learn a proper subset of the teacher’s knowledge). These two assumptions ensure a gradual decline in scientific understanding over the course of future generations.
These assumptions make it no surprise that deficiencies in science can only get worse over time. This only leaves two more questions: how did the deficiency begin, and how can it be fixed? I might take a bold but not entirely unreasonable guess that much of it can be attributed to what I previously called curriculum that is not “readily applicable to everyday situations”. Who is in charge of deciding this curriculum? The answer to this is: precisely those who originally learned this curriculum. So again, it is no surprise why nothing changes. Many point to issues like parenting, television, and other media as the problem. But each of these are easily traceable back to the fact that members of each of these components (parents and media affiliates) had a similar education anyway. This further reinforces the claim that the curriculum itself might need to change.
Now if we change the curriculum, the question of teachers’ ability to implement it arises. This can be addressed with changes in college curriculum that prepare emerging teachers for a new curriculum in the primary and secondary levels. This seems like it might be the first promising step to address the issue in the long run. In an era where governmental budget deficits are high, the appetite for further investment in education (itself a long-term yet critical investment) also seems to be on the decline. It’s no surprise that a fair amount of politicians’ value placed on scientific investment puts it on the back burner in a budget crisis given that a vast majority of them have no background in science to begin with. This is yet another reason that a change in curriculum seems more promising in my mind than further monetary investments in education. And since the payoffs for this will not be immediate, it is all the more reason to begin a change in curriculum now rather than later. If a scientific curriculum tied closer to reality and pragmatic obstacles implants itself in those who will become the future generations, the hope is that this will in turn resolve the future problems associated to the parents, media, and politicians.