Einsteinian gravity, of which Newtonian gravity is a part, is fraught with the problem of singularity that has been established as a theorem by Hawking and Penrose. The hypothesis that founds the basis of both Einsteinian and Newtonian theories of gravity is that bodies with unequal magnitudes of masses fall with the same acceleration under the gravity of a source object. Since, the Einstein’s equations is one of the assumptions that underlies the proof of the singularity theorem, therefore, the above hypothesis is implicitly one of the founding pillars of the same. In this work, I demonstrate how one can possibly write a non-singular theory of gravity which manifests that the above mentioned hypothesis is only valid in an approximate sense in the “large distance” scenario. To mention a specific instance, under the gravity of the earth, a 5 kg and a 500 kg fall with accelerations which differ by approximately \(113.148\times 10^{-32}\) meter/sec\(^2\) and the more massive object falls with less acceleration. Further, I demonstrate why the concept of gravitational field is not definable in the “small distance” regime which automatically justifies why the Einstein’s and Newton’s theories fail to provide any “small distance” analysis. In course of writing down this theory, I demonstrate why the continuum hypothesis as spelled out by Goedel, is undecidable. The theory has several aspects which provide the following realizations: (i) Descartes’ self-skepticism concerning exact representation of numbers by drawing lines (ii) Born’s wish of taking into account “natural uncertainty in all observations” while describing “a physical situation” by means of “real numbers” (iii) Klein’s vision of having “a fusion of arithmetic and geometry” where “a point is replaced by a small spot” (iv) Goedel’s assertion about “non-standard analysis, in some version” being “the analysis of the future”.

爱因斯坦引力是牛顿引力的一部分，它充满了奇点问题，奇点已被霍金和彭罗斯确立为定理。假设_爱因斯坦和牛顿引力理论的基础是，质量大小不等的物体在源物体的引力作用下以相同的加速度下落。由于爱因斯坦方程是奇点定理证明的基础假设之一，因此，上述假设隐含地是奇点定理的奠基支柱之一。在这项工作中，我展示了如何写出一种非奇异引力理论，表明上述假设仅在“远距离”场景下的近似意义上有效。举一个具体的例子，在地球引力作用下，5 公斤和 500 公斤的下落加速度相差大约\(113.148\times 10^{-32}\)米/秒\(^2\)更大质量的物体以更小的加速度下落。此外，我还论证了为什么引力场的概念在“小距离”范围内是不可定义的，这自动证明了为什么爱因斯坦和牛顿的理论无法提供任何“小距离”分析。在写下这个理论的过程中，我证明了为什么 Goedel 阐明的连续统假设是不可判定的。该理论有几个方面，提供了以下实现：