We study the pigeonhole principle for Σ 2 -definable injections with domain twice as large as the codomain, and the weak König lemma for ∆ 0 2 -definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of 2-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for Σ 2 -definable injections.

Hypertension presenting with metabolic alkalosis with hypokalemia is not uncommon. In this analysis, we present the management of such a scenario in white and black patients. Six patients (white¼4 [1 females, 2 males, age¼24-62]; blacks¼2 [1male, 1 female, age¼32, 40]) with hypokalemia (2.9-3.3 mEq/L), metabolic alkalosis (28-32 mm Eq/L) and hypertension (155-180/90-105 mmHg) were seen in hypertension clinic. All had a BMI of >32 and metabolic syndrome. None had clinical evidence of sleep apnea. All had been on adequate dosage of at least three BP meds (calcium channel blocker, ACE-I/ARB, metoprolol/carvedilol). All had a negative Doppler ultrasound for renal artery stenosis. Their serum aldosterone to renin ratio was 15-18 (normal¼less than 20). Urinary aldosterone in the six patients ranged from 10-12 ng/l (reference abnormal¼>14 ng/dl). Because of normal aldosterone to renin ration, CT scan was not obtained in these patients. Nonetheless, considering obesity and aldosterone production by adipocytes (high normal urinary aldosterone), treatment with spironolactone/eplerenone was initiated with normalization of blood pressure in 3/4 whites (<140/90 mmHg). The non-responder (white female age 62) was treated with amiloride with subsequent control of BP to <140/90 mmHg. Two black patients also responded better to amiloride compared to spironolactone/eplerenone bring them to the goal (<140/90 mmHg). While aldosterone is an important mediator of hypertension in obese individuals, over-activity of sodium channel in the distal tubule (ENaC) must be considered in the management uncontrolled hypertension with hypokalemic metabolic alkalosis, particularly in blacks. It has been suggested before that amiloride provides better BP control in resistance hypertension in blacks compared to white patients providing validity to ENaC over-activity. Genetic testing was not performed in our cohort to investigate the presence of Liddle's syndrome.

We examine the reverse-mathematical strength of several theorems in classical and effective model theory concerning first-order theories and their number of models. We prove that, among these, most are equivalent to one of the familiar systems RCA0, WKL0, or ACA0. We are led to a purely model-theoretic statement that implies WKL0 but refutes ACA0 over RCA0.

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We study the pigeonhole principle for Σ 2 -definable injections with domain twice as large as the codomain, and the weak König lemma for ∆ 0 2 -definable trees in which every level has at least half of the possible nodes. We show that the latter implies the existence of 2-random reals, and is conservative over the former. We also show that the former is strictly weaker than the usual pigeonhole principle for Σ 2 -definable injections.