I want to prove that:  Well ordering principle ⟺ Complete Induction.  I am interested in both directions of the implication. That is, that if every non-empty set contains a least element then if a property P holds for all naturals less than n-1 it must hold for n. And viceversa.  I want the proof to be classical (I suspect this is quite difficult to prove constructively)
Here is my attempt to prove the Well-Ordering Principle, i.e. that any non-empty subset of $\Bbb N$, the set of natural numbers, has a minimum element.  Proof. Suppose there exists a non-empty subset $S$ of $\Bbb N$ such that $S$ has NO minimum element. Define $A = \left\{n\in \Bbb N : (\forall s\in S)(n \leq s)\right\}$. It is obvious that $1\in A$. Suppose $n\in A$, then for each $s \in S$, there exists $q \in N$ such that $n + q = s$. Since $q \ge 1$, $n+1 \leq s$, for all $s\in S$. By Principle of mathematical induction, $A = \Bbb N$. Take any $s_0 ∈ S$, then $(\forall s\in S)(s_0 \leq s)$. (This contradicts that $S$ has no minimum element).  How do I prove the statement without invoking Mathematical Induction? Also, I read that the proof of Principle of Mathematical Induction makes use of Well-Ordering. Can it be proven independently of Well-Ordering too?
Polynomial contrasts for regression Profile picture cannot have transparent background Possible to have the default landing page of my Sharepoint Site be an applicaiton page? So basically when someone goes to:   http://mysite.com   I would like to show an Application Page that I have created, without the giant URL.  Example of the giant URL :  http://mysite.com/_layouts/ap/myAppPage.aspx   Is there any way for me to do this? Cards changing mana color What cards across all sets allow the utilization of mana, either colorless or otherwise, as colored mana of another color?  One early example with an unlimited  1:1 ratio:     Sunglasses of Urza   Continuous Artifact: White mana in your mana pool can also be used as red mana.   What other cards have similar abilities? So I have this word problem and I’m a bit confused about it. I have the answer and explanation but I still don’t understand: The half-life of carbon-14 is approximately 5730 years, while the half-life of carbon-12 is essentially infinite. If the ratio of carbon-14 to carbon-12 in a certain sample is 25% less than the normal ratio in nature, how old is the sample?     A. Less than 5730 years      B. Approximately 5730 years      C. Significantly greater than 5730 years, but less than 11460 years      D. Approximately 11460 years  Correct Answer: A Explanation: Because the half-life of carbon-12 is essentially infinite, a 25 percent decrease in the ratio of carbon-14 to carbon-12 means the same as a 25 percent decrease in the amount of carbon-14. If less than half of the carbon-14 has deteriorated, then less than one half-life has elapsed. Therefore, the sample is less than 5730 years old. Be careful with the wording here—the question states that the ratio is 25% less than the ratio in nature, not 25% of the ratio in nature, which would correspond to choice (D). How is the ratio of the carbon isotopes relevant to half-lives? What is the purpose of saying the half-life of an isotope is infinite? What is meant by “the normal ratio in nature”? Just based on the answer, it seems like the question said “25% of a carbon sample decayed, how old is this sample?” Obviously it’s younger than 5,730 years (the time for its first half life) because if only 25% decayed that means half a half-life has passed. I don’t see how isotopes and ratios would change this problem at all. Edit: Every answer here was very informative and helpful. It was tough to pick a best answer. I picked Cosma’s because it clicked with me. If you’re reading this and want to understand radiocarbon dating, look at every answer because there is some useful information in every answer How to output a counter with leading zeros? I can output the value of a counter in a LaTeX document via:  \setcounter{mycounter}{1} \arabic{mycounter}   Which inserts a 1. But how do I format the output by, for example, adding a leading zero if mycounter is less than 10 in a way similar to what  lets you do by specifying %02d as your format string? Can I see a list of gyms I left a pokemon at? I captured a gym today just before the servers went down. I am wondering if I lost my progress. Is there a way to see if I own a gym or not? Subscribe to a particular question by email (not RSS) Theorem  Let $\{a_n\}$ and $\{b_n\}$ be convergent real sequences. Assume that there exists a $N\in\mathbb{N}$ so  $a_n\le b_n$ (eq. 1)  for all $n\ge N$. Then  $\lim_{n\to\infty}a_n\le \lim_{n\to\infty}b_n$.  My attempt at proving  Let $a=\lim_{n\to\infty}a_n$ and $b=\lim_{n\to\infty}b_n$. $a$ and $b$ are real numbers because the limit of a sequence of real numbers is a real number (which is a corollary in my textbook). For proof by contradiction, assume  $b&lt;a$.  Let $\epsilon=\frac{a-b}{2}$. Then we can find $N_a,N_b\in\mathbb{N}$ so  $|a-a_n|\le\frac{\epsilon}{2}$ when $n\ge N_a$, (eq. 2)  $|b-b_n|\le\frac{\epsilon}{2}$ when $n\ge N_b$. (eq. 3)  In a textbook I found that I should do the following:  Choose a $n$ so $n\ge\max(N,N_a,N_b$), then eqs. (1-3) are satisfied. Then  $a-\epsilon\le a_n+\frac{\epsilon}{2}-\epsilon$  How do I get that last thing? Is it necessary to do a proof by contradiction or can one do a direct proof? My 5 year- old daughter is constantly defiant! My 5-year old daughter is very defiant.  It started getting bad about 6 months ago. She doesn't do what she is told, and any time she hears the word "no" she throws terrible, dramatic fits, screaming ridiculously loud, and in some cases hitting.  She refuses to make eye contact when I warn her to stop doing something even when I am down at her eye level and it takes multiple orders to finally get her to look at me. She constantly tells me no.  I feel like I constantly tell her no and that she feels this gives her the right to tell her father and I no.  I have tried time outs which are not effective at all; she only throws a fit the whole time, never seeming to run out of wind. Spankings are only effective until the shock of the action itself wears off and then she returns to the same behavior.  My husband and I recently took away ALL of her toys/ possessions being she is a pretty materialistic child (she loves getting toys and always wants something new). We told her that if she was good and listened and stopped telling us no and throwing fits, that after three days we would let her choose three boxes of toys to bring back in her room. This worked but as soon as she got those three boxes unpacked, we packed them back up because she went right back to the no's and fit-throwing within a few hours.  I am mainly the one disciplining her and I feel she is definitely the most defiant with me. I believe in explaining to her why things are not OK and why I tell her no, but I really don't feel it helps and I never feel as if she listens to what I tell her.  I am at my wits' end and feel like a failure of a mother. She is my first and only child, and the only grandchild on both sides of our family. She is an angel at school but home is completely different.  I need help figuring this out but nobody I talk to really has any advice for me. I am at the point where I am wondering if she or I need counseling because I feel out of options. It seems the .NET Regex.Replace method automatically replaces all matching occurrences.  I could provide a MatchEvaluator delegate that returns the matched string after the first replacement, rendering no change, but that sounds very inefficient to me.  What is the most efficient way to stop after the first replacement? Class object affecting other class object when thread.sleep- Python Connecting to Oracle database with ArcGIS for Desktop? Packet sniffing in Python (Windows) What is the best way to sniff network packets using Python?  I've heard from several places that the best module for this is a module called Scapy, unfortunately, it makes python.exe crash on my system. I would assume that it's just a problem with how I installed it, except that many other people have told me that it doesn't work particularly well on Windows. (If anyone is interested, I'm running Windows Vista, which might affect things).  Does anyone know of a better solution?  UPD:  After reading the answer telling me to install PyPcap, I messed around with it a bit and found out that Scapy, which I had tried using, was telling me to install PyPcap as well, except that it's a modified version for it's use. It was this modified PyPcap that was causing the problem, apparently, since the example in the answer also caused a hang.  I installed the original version of PyPcap (from Google's site), and Scapy started working fine (I didn't try many things, but at least it didn't crash as soon as I started sniffing). I sent a new defect ticket to the Scapy developers: , hope they can do something with it.  Anyways, just thought I'd let y'all know. Flawed proof that all positive integers are equal Give us better search-by-score syntax Significant figures vs. absolute error On , $N_A = 6.022\;141\;29 \times 10^{23} \text{ mol}^{-1}$ has 9 significant figures and a standard uncertainty of $0.000\;000\;27 \times 10^{23} \text{ mol}^{-1}$.  First, can it be written as the following?  $$ N_A = 6.022\;141\;29 \times 10^{23} \pm 0.000\;000\;27 \times 10^{23} \text{ mol}^{-1} $$ If yes, consider the following quote from  (emphasis added):     Whenever you make a measurement, the number of meaningful digits that you write down implies the error in the measurement. For example if you say that the length of an object is 0.428 m, you imply an uncertainty of about 0.001 m. To record this measurement as either 0.4 or 0.42819667 would imply that you only know it to 0.1 m in the first case or to 0.00000001 m in the second. You should only report as many significant figures as are consistent with the estimated error. The quantity 0.428 m is said to have three significant figures, that is, three digits that make sense in terms of the measurement. Notice that this has nothing to do with the "number of decimal places". The same measurement in centimeters would be 42.8 cm and still be a three significant figure number. The accepted convention is that only one uncertain digit is to be reported for a measurement. In the example if the estimated error is 0.02 m you would report a result of 0.43 ± 0.02 m, not 0.428 ± 0.02 m.   and the following quote from  (emphasis added):     The absolute error places bounds telling you that the actual value is expected to fall between x - D x and x + D x. The absolute error is an easy way to report the error and makes it easy to check that you are not reporting too many significant figures. For example, suppose you had a measurement that was reported to be 10.02 ± 0.3 m. Because the reported value is uncertain in the tenths place, it is wrong to report it to the hundredths place. It should be reported to the same decimal place as the absolute error: 10.0 ± 0.3 m. Another point is that the absolute error and the reported value must have the same units.   If there is such a convention (cf. Rice's lab guide) and it is wrong to report more significant figures than is warranted by the absolute error (cf. the joint project's guidelines), and NIST should have adhered to such convention and striven to be correct, is it true that NIST adheres and is correct because the standard uncertainty has two least significant digits (i.e., $27$)?  Had the standard uncertainty been $0.000\;000\;2\text{ mol}^{-1}$, would NIST have reported Avogadro's Number as $N_A = 6.022\;141\;3\times 10^{23}$ with 8 significant figures? Is the possessive of "one" spelled "ones" or "one's"? I've been confused about this as long as I can remember. Should it be:     One should do ones duty.   or     One should do one's duty.   I'm guessing it should be the latter. But that doesn't sit well with the possessive pronoun 'its'. For example:     It is its own purpose.   vs.     It is it's own purpose.   Here, the former seems clearly correct. How to to calculate the hash of an unordered set Suppose I have a set of elements, with known hash (e.g. SHA-2). How can I calculate the hash of the set? With it I mean an unordered set, so the order of elements is undefined and shall not play any role in determining the hash of the set.  In theory any commutative operation applied to the hashes should be fine, but I have the strong feeling that XORing all of them together is quite unsafe. An attacker could add an element to the set to produce a hash collision, although finding an element with that wanted hash is as hard as breaking the hash of the whole set.  Anything better solution than the trivial one of sorting the elements into a list, calculating their hashes and then calculating the hash of the hash list?  I think Merkle trees can't help here if they are based on some sort of ordering of the elements. How did Enrico Fermi calculate the classical Fermi Problem? From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude.  I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation.  See also: