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...even if natural numbers aren't. That is also true of every superset of the integers. Now we only know about integers. What x satisfies 2*x = 1? Start with 0 and 1, multiply by 2, and use ordering: 0 < 1 implies 2*0 < 2*1 giving 0 < 2 Where does 1 fit in? 0 < 1 < 2 Thus, x cannot be an...

...N <= F. Predicate function: if for function f, f(0) is true, and if for all a in N, f(a) implies f(S(a)), then for all a in N, f(a) is true. Addition: a + 0 = a a + S(b) = S(a + b) Multiplication: a*0 = a a*S(b) = (a*b) + a One can show that finite set cardinalities satisfy Peano's axioms.

What wind, currents and geography tell us about how people first settled Oceania 5200 BP - Taiwan (pause) 4000 BP - The Philippines Southward and eastward spread 3400 BP - Samoa and Tonga (pause) 1200 BP - Eastern Polynesia: Society Islands, Hawaii, Rapanui, New Zealand, visit to South America...

Linguistics locates the beginnings of the Austronesian expansion – with Indigenous seafaring people in eastern Taiwan Proto-Austronesian speakers traveled from South China to Taiwan around 5500 - 5000 BP (3500 - 3000 BCE), around the time of the Proto-Indo-European speakers. An offshoot...

Let's see The Numbers List - a big collection of words from 1 to 10 - also Numeral Systems of the World Quechuan: +Proto-Quechua - *suk - *ishkay - *kimsa - *çusku, *tawa - *pichqa - - *suqta - *qançis - *pusaq - *isqun - *çunka C Quechua (Ancash) - huk - ishkay - kima - chusku - pitsqa - -...

...evidence that Polynesians came from the west, the islands off of Southeast Asia, not the east, the west coast of South America. As to the word *kumala ~ *kumara for sweet potato (Ipomoea batatas), it traveled with the vegetable, making it a wander word. Entries for KUMALA.1 [PN] Sweet...

At 1000 AU, the gravitational-lens effect sizes: Neutron star: 1", Cygnus X-1: 4", Gaia BH3: 5", Sgr A*: 32', M87* 21d. At 1000 AU, the amount of perturbation of the Earth's orbit over a year: NS: ~ 10^(-9), Cyg X-1, GBH3: 3*10^(-8), Sgr A*: 4*10^(-3), M87*: (disruption) So it would be hard to...

A dead pulsar will be hard to observe with nongravitational effects, so I consider gravitational effects. These will have similar sizes for active pulsars and for white dwarfs, I must note, though those objects are easily observed nongravitationally. At 1000 AU distance, its gravitational-lens...

...PULSARS It states that the radio luminosities of pulsars range from 10^27 to 10^31 ergs/s, or 10^20 to 10^24 watts. The Sun's luminosity is 3.827*10^(26) watts. (PDF) Distances and other parameters for 1328 radio pulsars and Pulsar - pulsars are often observed over sizable fractions of our...

A Sunlike star would make itself very obvious once it reaches the distance of the nearest stars to the Solar System - it would be one of the brightest stars in the sky. List of nearest stars - Alpha Centauri: 1.33 pc, 4.34 ly, mag -0.27, Sirius: 2.67 pc, 8.71 ly, mag -1.46. At 1 pc, the Sun...

...much smaller than the largest rocky planets elsewhere: the Earth, our home planet. Its brightness from reflected sunlight would be magnitude -4 + 5*log10(dsp) + 5*log10(dep) for Sun-planet distance dsp and Earth-planet distance dep in AU's (size of Earth orbit). For dsp >> 1 AU, that reduces...

...hole approached the Solar System, what would we see of it as it approaches? Its gravitational-lens effect for some source s would have size sqrt(R*Dr) where R is the black-hole radius and Dr is the reduced total distance to the source: Ds*Do/(Ds+Do) where Ds is the distance from the lenser...

The Boston Tea Party?

Are Israeli civilians human shields because IDF soldiers live among them? Is Tel Aviv a human-shield city because the headquarters of the IDF is in it? That's the sort of arguments that Israel apologists are making about Hamas and Gazans. If the Israel-Hezbollah war becomes much hotter...

List of proposed Solar System spacecraft - mostly missions to the Moon, Mars, and Venus, but there are some to various asteroids.

What campaign-finance laws? What waiving of ballot-access laws? What debate exclusion? Presidential-debate hosts may not want to host marginal candidates, vanity-run candidates, and candidates who are not very serious. I suggest forgetting about the Presidency and focusing on downballot races...

The Founders seemed to want a no-party system, but once the government they created went into action, its participants split up into two parties: the Federalists and the Democratic-Republicans, the First Party System. The Feds were destroyed by the DR's, who co-opted much of their platform, and...

Apple has released its Apple M4 for its iPad Pro (7th generation). It's overall similar to its predecessors, but it's a little faster. It also has support for hardware ray tracing, good for improved real-time lighting and shadows.

You searched for flow batteries - CleanTechnica Vanadium Flow Batteries Are Coming For Your Gas Power Plants - "The US Department of Energy has tapped six sites to host new vanadium flow batteries, aiming to replace fossil energy with renewables." Here’s the Top 10 List of Flow Battery...

Cleantech News — #1 In EV, Solar, Wind, Tesla News - a very busy site. Sodium-ion batteries are getting attention as an alternative to lithium-ion ones, because Na is MUCH more common than Li. You searched for sodium-ion - CleanTechnica US Gets First Sodium-Ion Battery Factory Is that...